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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:09:28 -0700 |
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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:09:28 -0700 |
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You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: Orders of Infinity % +% The 'Infinitärcalcül' of Paul Du Bois-Reymond % +% % +% Author: Godfrey Harold Hardy % +% % +% Release Date: November 25, 2011 [EBook #38079] % +% Most recently updated: June 11, 2021 % +% % +% Language: English % +% % +% Character set encoding: UTF-8 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK ORDERS OF INFINITY *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{38079} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: Additional mathematical symbols. 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You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Orders of Infinity + The 'Infinitärcalcül' of Paul Du Bois-Reymond + +Author: Godfrey Harold Hardy + +Release Date: November 25, 2011 [EBook #38079] +Most recently updated: June 11, 2021 + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK ORDERS OF INFINITY *** +\end{PGtext} +\end{minipage} +\end{center} +\clearpage + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Brenda Lewis and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive/Canadian Libraries) +\end{PGtext} +\end{minipage} +\vfill +\TranscribersNote{\TransNoteText} +\end{center} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +%% -----File: 001.png---Folio xx------- +\cleardoublepage +\pagenumbering{roman} +\null\vfill +\begin{center} +\Titlefont{Cambridge Tracts in Mathematics \\[12pt] +and Mathematical Physics} +\bigskip + +\textsc{General Editors} +\medskip + +J. G. LEATHEM, M.A. \\ +E. T. WHITTAKER, M.A., F.R.S. +\vfill + +\Titlefont{No.\ 12 \\[24pt] +ORDERS OF INFINITY} +\end{center} +%% -----File: 002.png---Folio xx------- +\clearpage +\begin{center} +\large +CAMBRIDGE UNIVERSITY PRESS \\ +\textgoth{London}: FETTER LANE, E.C. \\ +C. F. CLAY, \textsc{Manager} +\bigskip + +\Graphic[png]{1.25in}{cups} +\bigskip + +\normalsize +\textgoth{Edinburgh}: 100, PRINCES STREET \\ +\textgoth{Berlin}: A. ASHER AND CO. \\ +\textgoth{Leipzig}: F. A. BROCKHAUS \\ +\textgoth{New York}: G. P. PUTNAM'S SONS \\ +\textgoth{Bombay and Calcutta}: MACMILLAN AND CO., \textsc{Ltd.} +\vfill + +\textit{All rights reserved} +\end{center} +%% -----File: 003.png---Folio xx------- +\clearpage +\begin{center} +\Titlefont{\Huge ORDERS OF INFINITY} +\bigskip + +{\large THE `INFINITÄRCALCÃœL' OF \\[8pt] +PAUL DU BOIS-REYMOND} +\vfill +\vfill + +by +\bigskip + +G. H. HARDY, M.A., F.R.S. \\ +\medskip + +{\small Fellow and Lecturer of Trinity College, Cambridge} +\vfill +\vfill +\vfill + +{\large Cambridge: \\ +at the University Press + +1910} +\end{center} +%% -----File: 004.png---Folio xx------- +\clearpage +\null\vfill +\begin{center} +\textgoth{Cambridge}: +\medskip + +\footnotesize +PRINTED BY JOHN CLAY, M.A. +\medskip + +AT THE UNIVERSITY PRESS +\end{center} +\vfill +%% -----File: 005.png---Folio xx------- + +\Preface + +\First{The} ideas of Du~Bois-Reymond's \textit{Infinitärcalcül} are of great and +growing importance in all branches of the theory of functions. +With the particular system of notation that he invented, it is, no +doubt, quite possible to dispense; but it can hardly be denied that +the notation is exceedingly useful, being clear, concise, and expressive +in a very high degree. In any case Du~Bois-Reymond was a mathematician +of such power and originality that it would be a great pity if +so much of his best work were allowed to be forgotten. + +There is, in Du~Bois-Reymond's original memoirs, a good deal that +would not be accepted as conclusive by modern analysts. He is also +at times exceedingly obscure; his work would beyond doubt have +attracted much more attention had it not been for the somewhat +repugnant garb in which he was unfortunately wont to clothe his most +valuable ideas. I have therefore attempted, in the following pages, +to bring the \textit{Infinitärcalcül} up to date, stating explicitly and proving +carefully a number of general theorems the truth of which Du~Bois-Reymond +seems to have tacitly assumed---I may instance in particular +the theorem of~\Ref{iii.}{§\;2}. + +I have to thank Messrs J.~E. Littlewood and G.~N. Watson for +their kindness in reading the proof-sheets, and Mr J.~Jackson for the +numerical results contained in Appendix~III\@. + +\Signature{G. H. H.} +{\textsc{Trinity College},} +{\textit{April}, 1910.} +%% -----File: 006.png---Folio xx------- +%% -----File: 007.png---Folio xx------- +\Contents + +\PageLine + +\ToCChap{I.}{Introduction}{1} + +\ToCChap{II.}{Scales of infinity in general}{7} + +\ToCChap{III.}{Logarithmico-exponential scales}{16} + +\ToCChap{IV.}{Special problems connected with logarithmico-exponential +scales}{21} + +\ToCChap{V.}{Functions which do not conform to any logarithmico-exponential +scale}{26} + +\ToCChap{VI.}{Differentiation and integration}{36} + +\ToCChap{VII.}{Some developments of Du Bois-Reymond's \textit{Infinitärcalcül}}{41} + +% Prints heading "Appendix I." +\ToCApp{I.}{General Bibliography}{47} + +\ToCApp{II.}{A sketch of some applications, with references}{48} + +\ToCApp{III.}{Some numerical results}{58} + +%% -----File: 008.png---Folio xx------- +%% -----File: 009.png---Folio 1------- +\MainMatter + +\Chapter{I.}{Introduction.} + +\Paragraph{1.} \First{The} notions of the `order of greatness' or `order of smallness' +of a function~$f(n)$ of a positive integral variable~$n$, when $n$~is `large,' +or of a function~$f(x)$ of a continuous variable~$x$, when $x$~is `large' or +`small' or `nearly equal to~$a$,' are of the greatest importance even in +the most elementary stages of mathematical analysis.\footnote + {See, for instance, my \textit{Course of pure mathematics}, pp.~168~\textit{et seq.}, 183~\textit{et seq.}, + 344~\textit{et seq.}, 350.} +The student +soon learns that as $x$~tends to infinity ($x \to \infty$) then also $x^{2} \to \infty$, and +moreover that $x^{2}$~tends to infinity \emph{more rapidly than~$x$}, \ie\ that the +ratio~$x^{2}/x$ tends to infinity as well; and that $x^{3}$~tends to infinity more +rapidly than~$x^{2}$, and so on indefinitely: and it is not long before he +begins to appreciate the idea of a `scale of infinity'~$(x^{n})$ formed by the +functions $x$,~$x^{2}$, $x^{3}$,~\dots, $x^{n}$,~\dots. This scale he may supplement and to +some extent complete by the interpolation of fractional powers of~$x$, +and, when he is familiar with the elements of the theory of the +logarithmic and exponential functions, of irrational powers: and so he +obtains a scale~$(x^{\alpha})$, where $\alpha$~is any positive number, formed by all +possible positive powers of~$x$. He then learns that there are functions +whose rates of increase cannot be measured by any of the functions of +this scale: that $\log x$, for example, tends to infinity more slowly, and $e^{x}$ +more rapidly, than \emph{any} power of~$x$; and that $x/(\log x)$ tends to infinity +more slowly than~$x$, but more rapidly than any power of~$x$ less than +the first. + +As we proceed further in analysis, and come into contact with its +most modern developments, such as the theory of Fourier's series, +the theory of integral functions, or the theory of singular points of +analytic functions, the importance of these ideas becomes greater and +%% -----File: 010.png---Folio 2------- +greater. It is the systematic study of them, the investigation of +general theorems concerning them and ready methods of handling +them, that is the subject of Paul du~Bois-Reymond's \textit{Infinitärcalcül} +or `calculus of infinities.' + +\Paragraph{2.} The notion of the `order' or the `rate of increase' of a function +is essentially a relative one. If we wish to say that `the rate of +increase of~$f(x)$ is so and so' all we can say is that it is greater than, +equal to, or less than that of some other function~$\phi(x)$. + +Let us suppose that $f$~and~$\phi$ are two functions of the continuous +variable~$x$, defined for all values of~$x$ greater than a given value~$x_{0}$. +Let us suppose further that $f$~and~$\phi$ are positive, continuous, and +steadily increasing functions which tend to infinity with~$x$; and let us +consider the ratio~$f/\phi$. We must distinguish four cases: + +\Item{(i)} If $f/\phi \to \infty$ with~$x$, we shall say that the rate of increase, or +simply the \emph{increase}, of~$f$ is greater than that of~$\phi$, and shall write +\[ +f \cgt \phi. +\] + +\Item{(ii)} If $f/\phi \to 0$, we shall say that the increase of~$f$ is less than that +of~$\phi$, and write +\[ +f \clt \phi. +\] + +\Item{(iii)} If $f/\phi$ remains, for all values of~$x$ however large, between two +fixed positive numbers $\delta$,~$\Delta$, so that $0 < \delta < f/\phi < \Delta$, we shall say that +the increase of~$f$ is equal to that of~$\phi$, and write +\[ +f \ceq \phi. +\] + +It may happen, in this case, that $f/\phi$ actually tends to a definite +limit. If this is so, we shall write +\[ +f \ceqq \phi. +\] + +Finally, if this limit is \emph{unity}, we shall write +\[ +f \sim \phi. +\] + +When we can compare the increase of~$f$ with that of some standard +function~$\phi$ by means of a relation of the type $f \ceq \phi$, we shall say that +$\phi$~\emph{measures}, or simply \emph{is}, the increase of~$f$. Thus we shall say that +the increase of~$2x^{2} + x + 3$ is~$x^{2}$. + +It usually happens in applications that $f/\phi$~is monotonic (\ie\ +steadily increasing or steadily decreasing) as well as $f$~and~$\phi$ themselves. +It is clear that in this case $f/\phi$ must tend to infinity, or zero, +or to a positive limit: so that one of the three cases indicated above +%% -----File: 011.png---Folio 3------- +must occur, and we must have $f \cgt \phi$ or $f \clt \phi$ or $f \ceqq \phi$ (not merely +$f \ceq \phi$). We shall see in a moment that this is not true in general. + +\Item{(iv)} It may happen that $f/\phi$ neither tends to infinity nor to zero, +nor remains between fixed positive limits. + +\begin{Remark} +Suppose, for example, that $\phi_{1}$,~$\phi_{2}$ are two continuous and increasing +functions such that $\phi_{1} \cgt \phi_{2}$. A glance at the +figure (\Fig{1}) will probably show with sufficient +% [Illustration: Fig. 1.] +\Figure[0.45\textwidth]{1}{011} +clearness how we can construct, by means of a +`staircase' of straight or curved lines, running +backwards and forwards between the graphs of +$\phi_{1}$~and~$\phi_{2}$, the graph of a steadily increasing +function~$f$ such that $f = \phi_{1}$ for $x = x_{1}$, $x_{3}$,~\dots\ and +$f = \phi_{2}$ for $x = x_{2}$, $x_{4}$,~\dots. Then $f/\phi_{1} = 1$ for +%[** TN: Next line displayed in the original] +$x = x_{1}$, $x_{3}$,~\dots, +but assumes for $x = x_{2}$, $x_{4}$,~\dots\ values which +decrease beyond all limit; while $f/\phi_{2} = 1$ +for $x = x_{2}$, $x_{4}$,~\dots, but assumes for $x = x_{1}$, $x_{3}$,~\dots\ +values which increase beyond all limit; and $f/\phi$, +where $\phi$~is a function such that $\phi_{1} \cgt \phi \cgt \phi_{2}$, +as \eg\ $\phi = \sqrt{\phi_{1} \phi_{2}}$, assumes both values which +increase beyond all limit and values which +decrease beyond all limit. + +Later on (\Ref{v.}{§\;3}) we shall meet with cases of this kind in which the +functions are defined by explicit analytical formulae. +\end{Remark} + +\Paragraph{3.} If a positive constant~$\delta$ can be found such that $f > \delta \phi$ for all +sufficiently large values of~$x$, we shall write +\[ +f \cgeq \phi; +\] +and if a positive constant~$\Delta$ can be found such that $f < \Delta \phi$ for all +sufficiently large values of~$x$, we shall write +\[ +f \cleq \phi. +\] +If $f \cgeq \phi$ and $f \cleq \phi$, then $f \ceq \phi$. + +It is however important to observe (i)~that $f \cgeq \phi$ is not logically +equivalent to the negation of $f \clt \phi$\footnote + {The relations $f \cgeq \phi$, $f \clt \phi$ are mutually exclusive but not exhaustive: $f \cgeq \phi$ + implies the negation of $f \clt \phi$, but the converse is not true.} +and (ii)~that it is not logically +equivalent to the alternative `\emph{$f \cgt \phi$ or $f \ceq \phi$}.' Thus, in the example +discussed at the end of~§\;2, $\phi_{1} \cgeq f \cgeq \phi_{2}$, but no one of the relations +$\phi_{1} \cgt f$, etc.\ holds. If however we know that one of the relations +$f \cgt \phi$, $f \ceq \phi$, $f \clt \phi$ \emph{must} hold, then these various assertions \emph{are} +logically equivalent. +%% -----File: 012.png---Folio 4------- + +The reader will be able to prove without difficulty that the symbols +$\cgt$,~$\ceq$,~$\clt$ satisfy the following theorems. +\begin{align*} +&\text{If $f \cgt \phi$, $\phi \cgeq \psi$, then $f \cgt \psi$.} \\ +&\text{If $f \cgeq \phi$, $\phi \cgt \psi$, then $f \cgt \psi$.} \\ +&\text{If $f \cgeq \phi$, $\phi \cgeq \psi$, then $f \cgeq \psi$.} \\ +&\text{If $f \ceq \phi$, $\phi \ceq \psi$, then $f \ceq \psi$.} +\displaybreak[1] \\[6pt] +&\text{If $f \cgeq \phi$, then $f + \phi \ceq f$.} \\ +&\text{If $f \cgt \phi$, then $f - \phi \ceq f$.} +\displaybreak[1] \\[6pt] +&\text{If $f \cgt \phi$, $f_{1} \cgt \phi_{1}$, then $f + f_{1} \cgt \phi + \phi_{1}$.} \\ +&\text{If $f \cgt \phi$, $f_{1} \ceq \phi_{1}$, then $f + f_{1} \cgeq \phi + \phi_{1}$.} \\ +&\text{If $f \ceq \phi$, $f_{1} \ceq \phi_{1}$, then $f + f_{1} \ceq \phi + \phi_{1}$.} +\displaybreak[1] \\[6pt] +&\text{If $f \cgt \phi$, $f_{1} \cgeq \phi_{1}$, then $ff_{1} \cgt \phi \phi_{1}$.} \\ +&\text{If $f \ceq \phi$, $f_{1} \ceq \phi_{1}$, then $ff_{1} \ceq \phi \phi_{1}$.} +\end{align*} + +Many other obvious results of the same character might be stated, +but these seem the most important. The reader will find it instructive +to state for himself a series of similar theorems involving also the +symbols $\ceqq$~and~$\sim$. + +\Paragraph{4.} So far we have supposed that the functions considered all tend +to infinity with~$x$. There is nothing to prevent us from including also +the case in which $f$~or~$\phi$ tends steadily to zero, or to a limit other than +zero. Thus we may write $x \cgt 1$, or $x \cgt 1/x$, or $1/x \cgt 1/x^{2}$. Bearing +this in mind the reader should frame a series of theorems similar to +those of~§\;3 but having reference to \emph{quotients} instead of to sums or +products. + +It is also convenient to extend our definitions so as to apply to +\emph{negative} functions which tend steadily to~$-\infty$ or to~$0$ or to some other +limit. In such cases we make no distinction, when using the symbols +$\cgt$,~$\clt$, $\ceq$,~$\ceqq$, between the function and its modulus: thus we write +$-x \clt -x^{2}$ or $-1/x \clt 1$, meaning thereby exactly the same as by +$x \clt x^{2}$ or $1/x \clt 1$. But $f \sim \phi$ is of course to be interpreted as a +statement about the actual functions and not about their moduli. + +It will be well to state at this point, once for all, that all functions +referred to in this tract, from here onwards, are to be understood, +unless the contrary is expressly stated or obviously implied, to be +positive, continuous, and monotonic, increasing of course if they tend +to~$\infty$, and decreasing if they tend to~$0$. But it is sometimes convenient +%% -----File: 013.png---Folio 5------- +to use our symbols even when this is not true of all the +functions concerned; to write, for example, +\[ +1 + \sin x \clt x, \qquad +x^{2} \cgt x\sin x, +\] +meaning by the first formula simply that $|1 + \sin x|/x \to 0$. This +kind of use may clearly be extended even to complex functions +(\eg~$e^{ix} \clt x$). + +Again, we have so far confined our attention to functions of a +continuous variable~$x$ which tends to~$+\infty$. This case includes that +which is perhaps even more important in applications, that of functions +of the positive integral variable~$n$: we have only to disregard values of~$x$ +other than integral values. Thus $n! \cgt n^{2}$, $-1/n \clt n$. + +Finally, by putting $x = -y$, $x = 1/y$, or $x = 1/(y - a)$, we are led to +consider functions of a continuous variable~$y$ which tends to~$-\infty$ or~$0$ +or~$a$: the reader will find no difficulty in extending the considerations +which precede to cases such as these. + +In what follows we shall generally state and prove our theorems +only for the case with which we started, that of indefinitely increasing +functions of an indefinitely increasing continuous variable, and shall +leave to the reader the task of formulating the corresponding theorems +for the other cases. We shall in fact always adopt this course, except +on the rare occasions when there is some essential difference between +different cases. + +\Paragraph{5.} There are some other symbols which we shall sometimes find it +convenient to use in special senses. + +By +\[ +O(\phi) +\] +we shall denote a function~$f$, otherwise unspecified, but such that +\[ +|f| < K\phi, +\] +where $K$~is a positive constant, and $\phi$~a positive function of~$x$: this +notation is due to Landau. Thus +\[ +x + 1 = O(x), \qquad +x = O(x^{2}), \qquad +\sin x = O(1). +\] + +We shall follow Borel in using the same letter~$K$ in a whole series +of inequalities to denote a positive constant, not necessarily the same +in all inequalities where it occurs. Thus +\[ +\sin x < K, \qquad +2x + 1 < Kx, \qquad +x^{m} < Ke^{x}. +\] +{\Loosen If we use~$K$ thus in any finite number of inequalities which (like the +first two above) do not involve any variables other than~$x$, or whatever +other variable we are primarily considering, then all the values of~$K$ lie +%% -----File: 014.png---Folio 6------- +between certain absolutely fixed limits $K_{1}$~and~$K_{2}$ (thus $K_{1}$~might be +$10^{-10}$ and $K_{2}$~be~$10^{10}$). In this case all the~$K$'s satisfy $0 < K_{1} < K < K_{2}$, +and every relation $f < K\phi$ might be replaced by $f < K_{2}\phi$, and every +relation $f > K\phi$ by $f > K_{1}\phi$. But we shall also have occasion to use $K$ +in equalities which (like the third above) involve a parameter (here~$m$). +In this case $K$, though independent of~$x$, is a function of~$m$. Suppose +that $\alpha$,~$\beta$,~\dots\ are all the parameters which occur in this way in this +tract. Then if we give any special system of values to $\alpha$,~$\beta$,~\dots, we +can determine $K_{1}$,~$K_{2}$ as above. Thus all our $K$'s satisfy} +\[ +0 < K_{1}(\alpha, \beta, \dots) < K < K_{2}(\alpha, \beta, \dots), +\] +where $K_{1}$,~$K_{2}$ are positive functions of $\alpha$,~$\beta$,~\dots\ defined for any permissible +set of values of those parameters. But $K_{1}$~has zero for its +lower limit; by choosing $\alpha$,~$\beta$,~\dots\ appropriately we can make~$K_{1}$ as +small as we please---and, of course, $K_{2}$~as large as we please.\footnote + {I am indebted to Mr~Littlewood for the substance of these remarks.} + +It is clear that the three assertions +\[ +f = O(\phi), \qquad +|f| < K\phi, \qquad +f \cleq \phi +\] +are precisely equivalent to one another. + +When a function~$f$ possesses any property for all values of~$x$ greater +than some definite value (this value of course depending on the nature +of the particular property) we shall say that $f$~possesses the property +for $x > x_{0}$. Thus +\[ +x > 100 \quad (x > x_{0}), \qquad +e^{x} > 100 x^{2} \quad (x > x_{0}). +\] + +We shall use $\delta$ to denote an arbitrarily small but fixed positive +number, and $\Delta$~to denote an arbitrarily great but likewise fixed positive +number. Thus +\[ +f < \delta\phi \quad (x > x_{0}) +\] +means `however small~$\delta$, we can find~$x_{0}$ so that $f < \delta\phi$ for $x > x_{0}$,' +\ie\ means the same as $f \clt \phi$; and $\phi > \Delta f\ (x > x_{0})$ means the same: +and +\[ +(\log x)^{\Delta} \clt x^{\delta} +\] +means `any power of~$\log x$, however great, tends to infinity more +slowly than any positive power of~$x$, however small.' + +Finally, we denote by~$\epsilon$ a function (of a variable or variables +indicated by the context or by a suffix) whose limit is zero when the +variable or variables are made to tend to infinity or to their limits +in the way we happen to be considering. Thus +\[ +f = \phi(1 + \epsilon), \qquad +f \sim \phi +\] +are equivalent to one another. +%% -----File: 015.png---Folio 7------- + +In order to become familiar with the use of the symbols defined in the +preceding sections the reader is advised to verify the following relations; in +them $P_{m}(x)$,~$Q_{n}(x)$ denote polynomials whose degrees are $m$~and~$n$ and whose +leading coefficients are positive: +\begin{gather*} +P_{m}(x) \cgt Q_{n}(x) \quad (m > n), \qquad + P_{m}(x) \ceqq Q_{n}(x) \quad (m = n), \\ +P_{m}(x) \ceqq x^{m}, \qquad + P_{m}(x)/Q_{n}(x) \ceqq x^{m-n}, +\displaybreak[1] \\[6pt] +\sqrt{ax^{2} + 2bx + c} \ceqq x \quad (a > 0), \qquad + \sqrt{x + a} \sim \sqrt{x}, \\ +\sqrt{x + a} - \sqrt{x} \sim a/2\sqrt{x}, \qquad + \sqrt{x + a} - \sqrt{x} = O(1/\sqrt{x}), +\displaybreak[1] \\[6pt] +e^{x} \cgt x^{\Delta}, \qquad + e^{x^{2}} \cgt e^{\Delta x}, \qquad + e^{e^{x}} \cgt e^{x^{\Delta}}, \\ +\log x \clt x^{\delta}, \quad + \log P_{m}(x) \ceqq \log Q_{n}(x), \quad + \log \log P_{m}(x) \sim \log \log Q_{n}(x), +\displaybreak[1] \\[6pt] +x + a\sin x \sim x, \qquad + x(a + \sin x) \ceq x\quad (a > 1), \\ +e^{a + \sin x} \ceq 1, \qquad + \cosh x \sim \sinh x \ceqq e^{x}, \\ +x^{m} = O(e^{\delta x}), \qquad + (\log x)/x = O(x^{\delta-1}), +\displaybreak[1] \\[6pt] +1 + \frac{1}{2} + \dots + \frac{1}{n} \cgt 1, \qquad + 1 + \frac{1}{2^{2}} + \dots + \frac{1}{n^{2}} \ceqq 1, \\ +1 + \frac{1}{2} + \dots + \frac{1}{n} \sim \log n, \qquad + 1 + \frac{1}{2} + \dots + \frac{1}{n} - \log n \ceqq 1, \\ +n! \clt n^{n}, \qquad + n! \cgt e^{\Delta n}, \qquad + n! = n^{n^{1+\epsilon}} = n^{n(1 + \epsilon)}, \\ +n! \sim n^{n + \frac{1}{2}} e^{-n} \sqrt{2\pi}, \qquad + n!\, (e/n)^{n} = (1 + \epsilon) \sqrt{2\pi n}, \\ +\int_{1}^{x} \frac{dt}{t} \cgt 1, \qquad + \int_{1}^{x} \frac{dt}{t} \sim \log x, \qquad + \int_{2}^{x} \frac{dt}{\log t} \sim \frac{x}{\log x}. +\end{gather*} + + +\Chapter{II.}{Scales of infinity in general.} + +\Paragraph{1.} \First{If} we start from a function~$\phi$, such that $\phi \cgt 1$, we can, in a +variety of ways, form a series of functions +\[ +\phi_{1} = \phi,\quad +\phi_{2},\quad +\phi_{3},\ \dots,\quad +\phi_{n},\ \dots +\] +such that the increase of each function is greater than that of its +predecessor. Such a sequence of functions we shall denote for shortness +by~$(\phi_{n})$. + +One obvious method is to take $\phi_{n} = \phi^{n}$. Another is as follows: +If $\phi \cgt x$, it is clear that +\[ +\phi\{\phi(x)\} / \phi(x) \to \infty, +\] +%% -----File: 016.png---Folio 8------- +and so $\phi_{2}(x) = \phi \phi(x) \cgt \phi(x)$; similarly $\phi_{3}(x) = \phi \phi_{2}(x) \cgt \phi_{2}(x)$, and +so on.\footnote + {For some results as to the increase of such iterated functions see \Ref{vii.}{§\;2~(vi)}.} + +Thus the first method, with $\phi = x$, gives the scale $x$,~$x^{2}$, $x^{3}$,~\dots\ or~$(x^{n})$; +the second, with $\phi = x^{2}$, gives the scale $x^{2}$,~$x^{4}$, $x^{8}$,~\dots\ or~$(x^{2^{n}})$. + +\begin{Remark} +These scales are \emph{enumerable} scales, formed by a simple progression of +functions. We can also, of course, by replacing the integral parameter~$n$ by +a continuous parameter~$\alpha$, define scales containing a non-enumerable +multiplicity of functions: the simplest is~$(x^{\alpha})$, where $\alpha$~is any positive number. +But such scales fill a subordinate \textit{rôle} in the theory. +\end{Remark} + +It is obvious that we can always insert a new term (and therefore, +of course, any number of new terms) in a scale at the beginning or +between any two terms: thus $\sqrt{\phi}$ (or $\phi^{\alpha}$, where $\alpha$~is any positive +number less than unity) has an increase less than that of any term +of the scale, and $\sqrt{\phi_{n} \phi_{n+1}}$ or $\phi_{n}^{\alpha} \phi_{n+1}^{1-\alpha}$ has an increase intermediate +between those of $\phi_{n}$~and~$\phi_{n+1}$. A less obvious and far more important +theorem is the following + +\begin{Result}[Theorem of Paul du~Bois-Reymond.] Given any ascending +scale of increasing functions~$\phi_{n}$, \ie\ a series of functions such that +$\phi_{1} \clt \phi_{2} \clt \phi_{3} \clt \dots$, we can always find a function~$f$ which increases +more rapidly than any function of the scale, \ie\ which satisfies the +relation $\phi_{n} \clt f$ for all values of~$n$. +\end{Result} + +In view of the fundamental importance of this theorem we shall +give two entirely different proofs. + +\Paragraph{2.} (i)~We know that $\phi_{n+1} \cgt \phi_{n}$ for all values of~$n$, but this, of +course, does not necessarily imply that $\phi_{n+1} \geq \phi_{n}$ for all values of $x$~and~$n$ +in question.\footnote + {$\phi_{n+1} \cgt \phi_{n}$ implies $\phi_{n+1} > \phi_{n}$ for sufficiently large values of~$x$, say for $x > x_{n}$. + But $x_{n}$ may tend to~$\infty$ with~$n$. Thus if $\phi_{n} = x^{n}/n!$ we have $x_{n} = n + 1$.} +We can, however, construct a new scale of +functions~$\psi_{n}$ such that + +\Item{(\textit{a})} $\psi_{n}$ is identical with~$\phi_{n}$ for all values of~$x$ from a certain value +$x_{n}$ onwards ($x_{n}$, of course, depending upon~$n$); + +\Item{(\textit{b})} $\psi_{n+1} \geq \psi_{n}$ for all values of $x$~and~$n$. + +For suppose that we have constructed such a scale up to its $n$th~term~$\psi_{n}$. +Then it is easy to see how to construct~$\psi_{n+1}$. Since +$\phi_{n+1} \cgt \phi_{n}$, $\phi_{n} \sim \psi_{n}$, it follows that $\phi_{n+1} \cgt \psi_{n}$, and so $\phi_{n+1} > \psi_{n}$ from a +certain value of~$x$ (say~$x_{n+1}$) onwards. For $x \geq x_{n+1}$ we take $\psi_{n+1} = \phi_{n+1}$. +For $x < x_{n+1}$ we give $\psi_{n+1}$ a value equal to the greater of the values of +%% -----File: 017.png---Folio 9------- +$\phi_{n+1}$,~$\psi_{n}$. Then it is obvious that $\psi_{n+1}$~satisfies the conditions (\textit{a})~and~(\textit{b}). + +Now let +\[ +f(n) = \psi_{n}(n). +\] +From $f(n)$ we can deduce a continuous and increasing function~$f(x)$, +such that +\[ +\psi_{n}(x) < f(x) < \psi_{n+1}(x) +\] +for $n < x < n + 1$, by joining the points~$(n, \psi_{n}(n))$ by straight lines or +suitably chosen arcs of curves. + +\begin{Remark} +It is perhaps worth while to call attention explicitly to a small point that +has sometimes been overlooked (see, \eg, +Borel, \textit{Leçons sur la théorie des fonctions}, +p.~114; \textit{Leçons sur les séries à termes positifs}, +p.~26). It is not always the case that the +use of straight lines will ensure +\[ +f(x) > \psi_{n}(x) +\] +for $x > n$ (see, for example, \Fig{2}, where +the dotted line represents an appropriate +arc). +\end{Remark} +% [Illustration: Fig. 2.] +\Figure{2}{017} + +Then +\[ +f/\psi_{n} > \psi_{n+1}/\psi_{n} +\] +for $x > n + 1$, and so $f \cgt \psi_{n}$; therefore +$f \cgt \phi_{n}$ and the theorem is proved. + +\begin{Remark} +{\Loosen The proof which precedes may be made +more general by taking $f(n) = \psi_{\lambda_{n}} (n)$, where +$\lambda_{n}$~is an integer depending upon~$n$ and +tending steadily to infinity with~$n$.} +\end{Remark} + +(ii)~The second proof of Du~Bois-Reymond's Theorem proceeds on +entirely different lines. We can always choose positive coefficients~$a_{n}$ +so that +\[ +f(x) = \sum_{1}^{\infty} a_{n}\psi_{n}(x) +\] +is convergent for all values of~$x$. This will certainly be the case, for +instance, if +\[ +1/a_{n} = \psi_{1}(1) \psi_{2}(2) \dots \psi_{n}(n). +\] +For then, if $\nu$~is any integer greater than~$x$, $\psi_{n}(x) < \psi_{n}(n)$ for $n \geqq \nu$, +and the series will certainly be convergent if +\[ +\sum_{\nu}^{\infty} \frac{1}{\psi_{1}(1) \psi_{2}(2) \dots \psi_{n-1}(n-1)} +\] +is convergent, as is obviously the case. +%% -----File: 018.png---Folio 10------- + +Also +\[ +f(x)/\psi_{n}(x) > a_{n+1}\psi_{n+1}(x)/\psi_{n}(x) \to \infty, +\] +so that $f \cgt \phi_{n}$ for all values of~$n$. + +\begin{Remark} +\Paragraph{3.} Suppose, \eg, that $\phi_{n} = x^{n}$. If we restrict ourselves to values of~$x$ +greater than~$1$, we may take $\psi_{n} = \phi_{n} = x^{n}$. The first method of construction +would naturally lead to +\[ +f = n^{n} = e^{n\log n}, +\] +or $f = (\lambda_{n})^{n}$, where $\lambda_{n}$~is defined as at the end of §\;2~(i), and each of these functions +has an increase greater than that of any power of~$n$. The second method +gives +\[ +f(x) = \sum_{1}^{\infty} \frac{x^{n}}{1^{1} 2^{2} 3^{3} \dots n^{n}}. +\] + +It is known\footnote + {\textit{Messenger of Mathematics,} vol.~34, p.~101.} +that when $x$~is large the order of magnitude of this function +is roughly the same as that of +\[ +e^{\frac{1}{2}(\log x)^{2}/\log\log x}. +\] + +{\Loosen As a matter of fact it is by no means necessary, in general, in order to +ensure the convergence of the series by which $f(x)$~is defined, to suppose that +$a_{n}$~decreases so rapidly. It is very generally sufficient to suppose $1/a_{n} = \phi_{n}(n)$: +this is always the case, for example, if $\phi_{n}(x) = \{\phi(x)\}^{n}$, as the series} +\[ +\sum \left\{\frac{\phi(x)}{\phi(n)}\right\}^{n} +\] +is always convergent. This choice of~$a_{n}$ would, when $\phi = x$, lead to +\[ +f(x) = \sum \left(\frac{x}{n}\right)^{n} + \sim \sqrt{\frac{2\pi x}{e}}\, e^{x/e}.\footnote + {Lindelöf, \textit{Acta Societatis Fennicae}, t.~31, p.~41; Le~Roy, \textit{Bulletin des Sciences + Mathématiques}, t.~24, p.~245.\PageLabel{10}} +\] + +But the simplest choice here is $1/a_{n} = n!$, when +\[ +f(x) = \sum \frac{x^{n}}{n!} = e^{x} - 1; +\] +it is naturally convenient to disregard the irrelevant term~$-1$. + +\Paragraph{4.} We can always suppose, if we please, that $f(x)$~is defined by a power +series $\sum a_{n}x^{n}$ convergent for all values of~$x$, in virtue of a theorem of Poincaré's\footnote + {\textit{American Journal of Mathematics}, vol.~14, p.~214.} +which is of sufficient intrinsic interest to deserve a formal statement and +proof. + +\begin{Result} +Given any continuous increasing function~$\phi(x)$, we can always find an +integral function~$f(x)$ \(\ie\ a function~$f(x)$ defined by a power series $\sum a_{n}x^{n}$ +convergent for all values of~$x$\) such that $f(x) \cgt \phi(x)$. +\end{Result} + +The following simple proof is due to Borel.\footnote + {\textit{Leçons sur les séries à termes positifs}, p.~27.} + +Let $\Phi(x)$ be any function (such as the square of~$\phi$) such that $\Phi \cgt \phi$. Take +%% -----File: 019.png---Folio 11------- +an increasing sequence of numbers~$a_{n}$ such that $a_{n} \to \infty$, and another sequence +of numbers~$b_{n}$ such that +\[ +a_{1} < b_{2} < a_{2} < b_{3} < a_{3} < \dots; +\] +and let +\[ +f(x) = \sum \left(\frac{x}{b_{n}}\right)^{\nu_{n}}, +\] +where $\nu_{n}$~is an integer and $\nu_{n+1} > \nu_{n}$. This series is convergent for all values +of~$x$; for the $n$th~root of the $n$th~term is, for sufficiently large values of~$n$, not +greater than~$x/b_{n}$, and so tends to zero. Now suppose $a_{n} \leqq x < a_{n+1}$; then +\[ +f(x) > \left(\frac{a_{n}}{b_{n}}\right)^{\nu_{n}}. +\] +Since $a_{n} > b_{n}$ we can suppose $\nu_{n}$~so chosen that (i)~$\nu_{n}$~is greater than any of +$\nu_{1}$,~$\nu_{2}$, \dots,~$\nu_{n-1}$ and (ii) +\[ +\left(\frac{a_{n}}{b_{n}}\right)^{\nu_{n}} > \Phi(a_{n+1}). +\] + +Then +\[ +f(x) > \Phi(a_{n+1}) > \Phi(x), +\] +and so $f \cgt \phi$. +\end{Remark} + +\Paragraph{5.} So far we have confined our attention to ascending scales, such +as $x$,~$x^{2}$, $x^{3}$,~\dots, $x^{n}$,~\dots\ or~$(x^{n})$; but it is obvious that we may consider +in a similar manner \emph{descending} scales such as $x$,~$\sqrt{x}$, $\sqrt[3]{x}$,~\dots, $\sqrt[n]{x}$,~\dots\ +or~$(\sqrt[n]{x})$. It is very generally (though not always) true that if $(\phi_{n})$~is +an ascending scale, and $\psi$~denotes the function inverse to~$\phi$, then +$(\psi_{n})$~is a descending scale. + +\begin{Remark} +If $\phi > \bar{\phi}$ for all values of~$x$ (or all values greater than some definite value), +then a glance at \Fig{3} is enough to show that if +$\psi$~and~$\bar{\psi}$ are the functions inverse to $\phi$~and~$\bar{\phi}$, +then $\psi < \bar{\psi}$ for all values of~$x$ (or all values +greater than some definite value). We have only +to remember that the graph of~$\psi$ may be obtained +from that of~$\phi$ by looking at the latter from a +different point of view (interchanging the \textit{rôles} of +$x$~and~$y$). But it is not true that $\phi \cgt \bar{\phi}$ involves +$\psi \clt \bar{\psi}$. Thus $e^{x} \cgt e^{x}/x$. The function inverse +to~$e^{x}$ is~$\log x$: the function inverse to~$e^{x}/x$ is +obtained by solving the equation $x = e^{y}/y$ with +respect to~$y$. This equation gives +\[ +y = \log x + \log y, +\] +and it is easy to see that $y \sim \log x$. +\end{Remark} +%[Illustration: Fig. 3.] +\Figure[0.4\textwidth]{3}{019} + +\begin{Result} +Given a scale of increasing functions~$\phi_{n}$ such that +\[ +\phi_{1} \cgt \phi_{2} \cgt \phi_{3} \cgt \dots \cgt 1, +\] +%% -----File: 020.png---Folio 12------- +we can find an increasing function~$f$ such that $\phi_{n} \cgt f \cgt 1$ for all values +of~$n$.\end{Result} The reader will find no difficulty in modifying the argument +of §\;2~(i) so as to establish this proposition. + +\Paragraph{6.} The following extensions of Du~Bois-Reymond's Theorem +(and the corresponding theorem for descending scales) are due to +Hadamard.\footnote + {\textit{Acta Mathematica}, t.~18, pp.~319 \textit{et seq.}} + +\begin{Result} +Given +\[ +\phi_{1} \clt \phi_{2} \clt \phi_{3} \clt \dots \clt \phi_{n} \clt \dots \clt \Phi, +\] +we can find $f$ so that $\phi_{n} \clt f \clt \Phi$ for all values of~$n$. +\end{Result} + +\begin{Result} +Given +\[ +\psi_{1} \cgt \psi_{2} \cgt \psi_{3} \cgt \dots \cgt \psi_{n} \cgt \dots \cgt \Psi, +\] +we can find $f$ so that $\psi_{n} \cgt f \cgt \Psi$ for all values of~$n$. +\end{Result} + +\begin{Result} +Given an ascending sequence~$(\phi_{n})$ and a descending sequence~$(\psi_{p})$ +such that $\phi_{n} \clt \psi_{p}$ for all values of $n$~and~$p$, we can find $f$ so that +\[ +\phi_{n} \clt f \clt \psi_{p} +\] +for all values of $n$~and~$p$. +\end{Result} + +To prove the first of these theorems we have only to observe that +\[ +\Phi/\phi_{1} \cgt \Phi/\phi_{2} \cgt \dots \cgt \Phi/\phi_{n} \cgt \dots \cgt 1, +\] +and to construct a function~$F$ (as we can in virtue of the theorem +of~§\;5) which tends to infinity more slowly than any of the functions~$\Phi/\phi_{n}$. +Then +\[ +f = \Phi/F +\] +is a function such as is required. Similarly for the second theorem. +The third is rather more difficult to prove. + +\begin{Remark} +In the first place, we may suppose that $\phi_{n+1} > \phi_{n}$ for all values of $x$~and~$n$: +for if this is not so we can modify the +definitions of the functions~$\phi_{n}$ as in §\;2~(i). +Similarly we may suppose $\psi_{p+1} < \psi_{p}$ for all +values of $x$~and~$p$. + +Secondly, we may suppose that, if $x$~is +fixed, $\phi_{n} \to \infty$ as $n \to \infty$, and $\psi_{p} \to 0$ as +$p \to \infty$. For if this is not true of the +functions given, we can replace them by +$H_{n}\phi_{n}$ and $K_{p}\psi_{p}$, where $(H_{n})$~is an increasing +sequence of constants, tending to~$\infty$ with~$n$, +and $(K_{p})$~a decreasing sequence of constants +whose limit as $p \to \infty$ is zero. +% [Illustration: Fig. 4.] +\Figure{4}{020} + +Since $\psi_{p} \cgt \phi_{n}$ but, for any given~$x$, $\psi_{p} < \phi_{n}$ +for sufficiently large values of~$n$, it is clear +(see \Fig{4}) that the curve $y = \psi_{p}$ intersects the curve $y = \phi_{n}$ for all sufficiently +large values of~$n$ (say for $n \geq n_{p}$). +%% -----File: 021.png---Folio 13------- + +At this point we shall, in order to avoid unessential detail, introduce a +restrictive hypothesis which can be avoided by a slight modification of the +argument,\footnote + {See Hadamard's original paper quoted above.} +but which does not seriously impair the generality of the result. +We shall assume that no curve $y = \psi_{p}$ intersects any curve $y = \phi_{n}$ in more +than one point; let us denote this point, if it exists, by~$P_{n, p}$. + +If $p$ is fixed, $P_{n, p}$~exists for $n > n_{p}$; similarly, if $n$~is fixed, $P_{n, p}$~exists +for $p > p_{n}$. And as either $n$~or~$p$ increases, so do both the ordinate or the +abscissa of~$P_{n, p}$. The curve~$\psi_{p}$ contains all the points~$P_{n, p}$ for which $p$~has +a fixed value: and $y = \phi_{n}$ contains all the points for which $n$~has a fixed value. + +It is clear that, in order to define a function~$f$ which tends to infinity +more rapidly than any~$\phi_{n}$ and less rapidly than any~$\psi_{p}$, all that we have to +do is to draw a curve, making everywhere a positive acute angle with each of +the axes of coordinates, and crossing all the curves $y = \phi_{n}$ from below to +above, and all the curves $y = \psi_{p}$ from above to below. + +Choose a positive integer~$N_{p}$, corresponding to each value of~$p$, such that +(i)~$N_{p} > n_{p}$ and (ii)~$N_{p} \to \infty$ as $p \to \infty$. Then $P_{N_{p}, p}$~exists for each value of~$p$. +And it is clear that we have only to join the points $P_{N_{1}, 1}$,~$P_{N_{2}, 2}$, $P_{N_{3}, 3}$,~\dots\ by +straight lines or other suitably chosen arcs of curves in order to obtain a +curve which fulfils our purpose. The theorem is therefore established. +\end{Remark} + +\Paragraph{7.} Some very interesting considerations relating to scales of +infinity have been developed by Pincherle.\PageLabel{13}\footnote + {\textit{Memorie della Accademia delle Scienze di Bologna} (ser.~4, t.~5, p.~739).} + +We have defined $f \cgt \phi$ to mean $f/\phi \to \infty$, or, what is the same +thing, +\[ +\log f - \log \phi \to \infty. +\Tag{(1)} +\] + +We might equally well have defined $f \cgt \phi$ to mean +\[ +F(f) - F(\phi) \to \infty, +\Tag{(2)} +\] +where $F(x)$~is any function which tends steadily to infinity with~$x$ +(\eg~$x$,~$e^{x}$). Let us say that if \Eq{(2)}~holds then +\[ +f \cgt \phi \quad (F), +\Tag{(3)} +\] +so that $f \cgt \phi$ is equivalent to $f \cgt \phi\ (\log x)$. Similarly we define +$f \clt \phi\ (F)$ to mean that $F(f) - F(\phi) \to -\infty$, and $f \ceq \phi\ (F)$ to +mean that $F(f) - F(\phi)$ remains between certain fixed limits. Thus +\begin{gather*} +x + \log x \ceq x, \qquad x + \log x \cgt x \quad (x), \\ +x + 1 \ceq x\quad (x), \qquad x + 1 \cgt x \quad (e^{x}), +\end{gather*} +since $e^{x+1} - e^{x} = (e - 1)e^{x} \to \infty$. +%% -----File: 022.png---Folio 14------- + +It is clear that the more rapid the increase of~$F$, the more likely +is it to discriminate between the rates of increase of two given +functions $f$~and~$\phi$. More precisely, \begin{Result}if +\[ +f \cgt \phi \quad (F), +\] +and $\bar{F} = FF_{1}$, where $F_{1}$~is any increasing function, then will +\[ +f \cgt \phi \quad (\bar{F}). +\] +\end{Result} + +For +\[ +\bar{F}(f) - \bar{F}(\phi) = F(f) F_{1}(f) - F(\phi) F_{1}(\phi) + > \{F(f) - F(\phi)\} F_{1}(\phi) \to \infty. +\] + +\Paragraph{8.} The substance of the following theorems is due in part to +Pincherle and in part to Du Bois-Reymond.\footnote + {Pincherle, \lc; Du~Bois-Reymond, \textit{Math.\ Annalen}, Bd.~8, S.~390 \textit{et seq.}} + +\begin{Result} +\Item{1.} However rapid the increase of~$f$, as compared with that of~$\phi$, +we can so choose~$F$ that $f \ceq \phi\ (F)$. +\end{Result} + +\begin{Result} +\Item{2.} {\Loosen If $f - \phi$ is positive for $x > x_{0}$, we can so choose~$F$ that +$f \cgt \phi\ (F)$.} +\end{Result} + +\begin{Result} +{\Loosen \Item{3.} If $f - \phi$ is monotonic and not negative for $x > x_{0}$, and +$f \ceq \phi\ (F)$, however great be the increase of~$F$, then $f = \phi$ from a +certain value of~$x$ onwards.} +\end{Result} + +\Item{(1)} If $f \cgt \phi$, we may regard~$f$ as an increasing function of~$\phi$, say +\[ +f = \theta(\phi), +\] +where $\theta(x) \cgt x$. We can choose a constant~$g$ greater than~$1$, and then +choose~$X$ so that $\theta(x) > gx$ for $x > X$. Let $a$~be any number greater +than~$X$, and let +\[ +a_{1} = \theta(a), \qquad +a_{2} = \theta(a_{1}), \qquad +a_{3} = \theta(a_{2}),\ \dots. +\] +Then $(a_{n})$~is an increasing sequence, and $a_{n} \to \infty$, since $a_{n} > g^{n}a$. + +We can now construct an increasing function~$F$ such that +\[ +F(a_{n}) = \tfrac{1}{2} nK, +\] +where $K$~is a constant. Then if $a_{\nu-1} \leqq x \leqq a_{\nu}$, $a_{\nu} \leqq \theta(x) \leqq a_{\nu+1}$, and +\[ +F\{\theta(x)\} - F(x) < F(a_{\nu+1}) - F(a_{\nu-1}) < K. +\] +Accordingly $F(f) - F(\phi)$ remains less than a constant, and so the +first theorem is established. + +\Item{(2)} Let $f - \phi = \lambda$, so that $\lambda > 0$. If $\lambda$, as $x$~increases, remains +greater than a constant~$K$, then +\[ +e^{f} - e^{\phi} > (e^{K} - 1)e^{\phi} \to \infty, +\] +so that we may take $F(x) = e^{x}$. +%% -----File: 023.png---Folio 15------- + +If it is not true that $\lambda \geqq K$, $\lambda$~assumes values less than any +assignable positive number, as $x \to \infty$. Let $\bar{\lambda}(x)$ be defined as the +lower limit of~$\lambda(\xi)$ for $\xi \leqq x$. Then $\bar{\lambda}$~tends steadily to zero as $x \to \infty$, +and $\bar{\lambda} \leqq \lambda$. We may also regard $\bar{\lambda}$ as a steadily decreasing function +of~$\phi$, say $\bar{\lambda} = \mu(\phi)$. + +Let $\varpi(\phi)$ be an increasing function of~$\phi$ such that $\varpi \cgt 1/\mu$, $\mu\varpi \cgt 1$. +Then if +\begin{gather*} +F = \int^{\phi} \varpi(t)\, dt,\\ +F(f) - F(\phi) = \int_{\phi}^{\phi + \lambda} \varpi\, dt + \geqq \int_{\phi}^{\phi + \mu(\phi)} \varpi\, dt + > \mu(\phi)\varpi(\phi) \cgt 1, +\end{gather*} +and $F(x)$~fulfils the requirement of theorem~2. The third theorem is +obviously a mere corollary of the second. + +\begin{Remark} +The reader will find it instructive to deduce or prove independently the +following three theorems, which are closely analogous to those which have +just been proved. + +\begin{Result} +\Item{1.} However great be the increase of~$f$ as compared with that of~$\phi$, we can +determine an increasing function~$F$ such that $F(f) \ceq F(\phi)$. +\end{Result} + +\begin{Result} +\Item{2.} If $f - \phi$ is positive for $x > x_{0}$, we can determine an increasing function~$F$ +such that $F(f) \cgt F(\phi)$. +\end{Result} + +\begin{Result} +\Item{3.} If $f - \phi$ is monotonic and not negative for $x > x_{0}$, and $F(f) \ceq F(\phi)$, +however great the increase of~$F$, then $f = \phi$ from a certain value of~$x$ onwards. +\end{Result} + +{\Loosen To these he may add the theorem (analogous to that proved at the end of~§\;7) +that \begin{Result}$f \cgt \phi$ involves $F(f) \cgt F(\phi)$ if $\log F(x)/\log x$ is an increasing +function\end{Result} (a condition which may for practical purposes be replaced by +$F \cgt x$).} + +\Paragraph{9.} Let us consider some examples of the theorems of the last paragraph. + +\Item{(i)} Let $f = x^{m}$ ($m > 1$) and $\phi = x$. Then, following the argument of §\;8~(1), +we have $\theta(\phi) = \phi^{m}$. We may take +\[ +a = e, \qquad +a_{1} = e^{m}, \qquad +a_{2} = e^{m^{2}},\ \dots, \qquad +a_{n} =e^{m^{n}},\ \dots, +\] +and we have to define~$F$ so that +\[ +F(e^{m^{n}}) = \tfrac{1}{2}nK. +\] +The most natural solution of this equation is +\[ +F(x) = K\log\log x/2\log m. +\] +And in fact +\[ +F(x^{m}) - F(x) = \frac{K}{2\log m}\{\log(m\log x) - \log\log x\} + = \tfrac{1}{2}K, +\] +so that $x^{m} \ceq x\ (F)$. +%% -----File: 024.png---Folio 16------- + +\Item{(ii)} Let $f = e^{x} + e^{-x}$, $\phi = e^{x}$. Following the argument of §\;8~(2), we find +\[ +\lambda = e^{-x} = \bar{\lambda}, \qquad +\mu(\phi) = 1/\phi, +\] +and we may take $\varpi(\phi) = \phi^{1+\alpha}$ ($\alpha > 0$). This makes $F$ a constant multiple of~$x^{2+\alpha}$, +and it is easy to verify that +\[ +(e^{x} + e^{-x})^{k} - e^{kx} \to \infty, +\] +if $k > 2$. + +\Item{(iii)} The relation $F(f) \ceq F(\phi)$ is equivalent to $f \ceq \phi\ (\log F)$. Using +the result of~(i) we see that $F(x^{m}) \ceq F(x)$ if $F \cleq \log x$. Similarly, using the +result of~(ii), we see that $F(e^{x} + e^{-x}) \cgt F(e^{x})$ if $F \cgeq e^{x^{k}}$ ($k > 2$). +\end{Remark} + +\Paragraph{10.} Before leaving this part of our subject, let us observe that all +of the substance of §§\;1--6 of this section may be extended to the case +in which our symbols $\cgt$,~etc., are defined by reference to an arbitrary +increasing function~$F$. We leave it as an exercise to the reader to +effect these extensions. + +\Chapter{III.}{Logarithmico-Exponential Scales.} + +\Paragraph{1.} \First{The} only scales of infinity that are of any practical importance +in analysis are those which may be constructed by means of the +logarithmic and exponential functions. + +We have already seen (\Ref{ii.}{§\;3}) that +\[ +e^{x} \cgt x^{n} +\] +for any value of~$n$ however great. From this it follows that +\[ +\log x \clt x^{1/n} +\] +for any value of $n$.\footnote + {It was pointed out above (\Ref{ii.}{§\;5}) that $\phi \cgt \bar{\phi}$ does not necessarily involve $\psi \clt \bar{\psi}$ + ($\psi$,~$\bar{\psi}$ being the functions inverse to $\phi$,~$\bar{\phi}$). But it does involve $\psi < \bar{\psi}$ for sufficiently + large values of~$x$, and therefore $\psi \cleq \bar{\psi}$. Hence $\phi \cgt \phi_{n}$ (for any~$n$) involves $\psi \cleq \psi_{n}$ + (for any~$n$) and therefore, if $(\psi_{n})$~is a descending scale, as is in this case obvious, + $\psi \clt \psi_{n}$ for any~$n$. For proofs of the relations $e^{x} \cgt x^{n}$, $\log x \clt x^{1/n}$, proceeding on + different lines, see my \textit{Course of pure mathematics}, pp.~345,~350.} + +It is easy to deduce that +\begin{gather*} +e^{e^{x}} \cgt e^{x^{n}}, \qquad +e^{e^{e^{x}}} \cgt e^{e^{x^{n}}},\ \dots, \\ +\log\log x \clt (\log x)^{1/n}, \qquad +\log\log\log x \clt (\log\log x)^{1/n},\ \dots. +\end{gather*} +%% -----File: 025.png---Folio 17------- + +The repeated logarithmic and exponential functions are so important +in this subject that it is worth while to adopt a notation for +them of a less cumbrous character. We shall write +\begin{alignat*}{3} +%[** TN: Unaligned in the original] +l_{1}x &\eqq lx \eqq \log x, \qquad& +l_{2}x &\eqq llx, \qquad& +l_{3}x &\eqq ll_{2}x,\ \dots,\\ +e_{1}x &\eqq ex \eqq e^{x}, \qquad& +e_{2}x &\eqq eex, \qquad& +e_{3}x &\eqq ee_{2}x,\ \dots. +\end{alignat*} + +It is easy, with the aid of these functions, to write down any +number of ascending scales, each containing only functions whose +increase is greater than that of any function in any preceding scale; +for example +\begin{gather*} +x,\quad x^{2},\ \dots,\quad x^{n},\ \dots;\qquad +e^{x},\quad e^{2x},\ \dots,\quad e^{nx},\ \dots; \\ +e^{x^{2}},\quad e^{x^{3}},\ \dots,\quad e^{x^{n}},\ \dots;\qquad +e_{2}x,\quad e_{3}x,\ \dots,\quad e_{n}x,\ \dots. +\end{gather*} + +In among the functions of these scales we can of course interpolate +new functions as freely as we like, using, for instance, such functions as +\[ +x^{\alpha} e^{\beta x^{\gamma} e^{\delta x^{\epsilon}}}, +\] +where $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$ are any positive numbers; and we can of course +construct non-enumerable (\Ref{ii.}{§\;1}) as well as enumerable scales. +Similarly we can construct any number of descending scales, each +composed of functions whose increase is less than that of any functions +in any preceding scale: for example +\[ +lx, \quad (lx)^{1/2}, \ \dots, \quad (lx)^{1/n},\ \dots; \qquad +l_{2}x, \quad l_{3}x, \ \dots, \quad l_{n}x,\ \dots. +\] + +Two special scales are of particularly fundamental importance; the +ascending scale +\[ +\LTag{(E)} +x, \quad ex, \quad e_{2}x, \quad e_{3}x, \ \dots, +\] +and the descending scale +\[ +\LTag{(L)} +x, \quad lx, \quad l_{2}x, \quad l_{3}x, \ \dots. +\] + +These scales mark the \emph{limits} of all logarithmic and exponential +scales: it is of course, in virtue of the general theorems of~\Ref{ii.}{}, possible +to define functions whose increase is more rapid than that of any~$e_{n}x$ +or slower than that of any~$l_{n}x$; but, as we shall see in a moment, +this is \emph{not} possible if we confine ourselves to functions defined by +a finite and explicit formula involving only the ordinary functional +symbols of elementary analysis. + +\Paragraph{2.} We define a \emph{logarithmico-exponential function} (shortly, an +\emph{$L$-function}) as a real one-valued function defined, for all values of~$x$ +greater than some definite value, by a finite combination of the +ordinary algebraical symbols (viz.\ $+$,~$-$, $×$,~$÷$,~$\sqrt[n]{}$) and the functional +symbols $\log(\dots)$ and $e^{(\dots)}$, operating on the variable~$x$ and on real +constants. +%% -----File: 026.png---Folio 18------- + +\begin{Remark} +It is to be observed that the result of working out the value of the +function, by substituting~$x$ in the formula defining it, is to be real at all +stages of the work. It is important to exclude such a function +\[ +\tfrac{1}{2}\{e^{\sqrt{-x^{2}}} + e^{-\sqrt{-x^{2}}}\}, +\] +which, with a suitable interpretation of the roots, is equal to~$\cos x$. +\end{Remark} + +\begin{Theorem} +Any $L$-function is ultimately continuous, of constant +sign, and monotonic, and, as $x \to \infty$, tends to~$\infty$, or to zero or to some +other definite limit. Further, if $f$~and~$\phi$ are $L$-functions, one or other +of the relations +\[ +f \cgt \phi, \qquad +f \ceqq \phi, \qquad +f \clt \phi +\] +holds between them. +\end{Theorem} + +We may classify $L$-functions as follows, by a method due to +Liouville.\footnote + {See my tract \textit{The integration of functions of a single variable} (No.~2 of this + series), pp.~5 \textit{et~seq.}, where references to Liouville's original memoirs are given.} +An $L$-function is of order zero if it is purely algebraical; +of order~$1$ if the functional symbols $l(\dots)$ and $e(\dots)$ which occur +in it bear only on algebraical functions; of order~$2$ if they bear only +on algebraical functions or $L$-functions of order~$1$; and so on. Thus +\[ +x^{x^{x}} = e^{\log x e^{x\log x}} +\] +is of order~$3$. As the results stated in the theorem are true of +algebraical functions, it is sufficient to prove that, if true of $L$-functions +of order $n - 1$, they are true of $L$-functions of order~$n$. + +Let us observe first that if $f$~and~$\phi$ are $L$-functions, so is~$f/\phi$. +Hence the last part of the theorem is a mere corollary of the first +part. Again, the derivative of an $L$-function of order~$n$ is an $L$-function +of order~$n$ (or less). Hence it is enough to prove that, if +the results stated are true of $L$-functions of order~$n - 1$, then an +$L$-function of order~$n$ is ultimately continuous and of constant sign, +\ie\ that it is continuous and cannot vanish for a series of values of~$x$ +increasing beyond limit. For, if this is true of any $L$-function of +order~$n$, it is true of the derivative of any such function; and therefore +the function itself is ultimately continuous and monotonic. + +Now any $L$-function of order~$n$ can be expressed in the form +\begin{align*} +f_{n} &= A\{e\phi_{n-1}^{(1)}, e\phi_{n-1}^{(2)}, \dots, e\phi_{n-1}^{(r)},\ + l\psi_{n-1}^{(1)}, \dots, l\psi_{n-1}^{(s)}, + \chi_{n-1}^{(1)}, \dots, \chi_{n-1}^{(t)}\}\\ + &= A\{z_{1}, z_{2}, \dots, z_{q}\}, +\end{align*} +say, where $q = r + s + t$, the functions with suffix~$n - 1$ are $L$-functions +of order~$n - 1$, and $A$~denotes an algebraical function: and there is +therefore an identical relation +\[ +F \eqq M_{0} f_{n}^{p} + M_{1} f_{n}^{p-1} + \dots + M_{p} = 0, +\] +%% -----File: 027.png---Folio 19------- +where the coefficients are polynomials in $z_{1}$,~$z_{2}$, \dots,~$z_{q}$. These polynomials +are comprised in the class of functions +\[ +M = \sum \rho_{n-1} e\sigma_{n-1} (l\tau_{n-1}^{(1)})^{\kappa_{1}} (l\tau_{n-1}^{(2)})^{\kappa_{2}} \dots (l\tau_{n-1}^{(h)})^{\kappa_{h}}, +\] +in which the $\kappa$'s are positive integers, the number of terms in the +summation is finite, and the functions with suffix~$n - 1$ are again +$L$-functions of order~$n - 1$. So also are +\[ +\frac{dM_{0}}{dx}, \quad +\frac{dM_{1}}{dx},\ \dots, \quad +\frac{dM_{p}}{dx}, +\] +and the discriminant of~$F$ \textit{qua} function of~$f_{n}$. + +Let us suppose our conclusions established in so far as relates to +functions of the type~$M$. Then it follows by a well known theorem\footnote + {If $F(x, y)$ is a function of $x$~and~$y$ which vanishes for $x = a$, $y = b$, and has + derivatives $\dfrac{\dd F}{\dd x}$,~$\dfrac{\dd F}{\dd y}$ continuous about~$(a, b)$, and if $\dfrac{\dd F}{\dd y}$~does not vanish for $x = a$, + $y = b$, then there is a unique continuous function~$y$ which is equal to~$b$ when $x = a$, + and satisfies the equation $F(x, y) = 0$ identically. See, \eg, W.~H.~Young, \textit{Proc.\ + Lond.\ Math.\ Soc.}, vol.~7, pp.~397 \textit{et~seq.}} +that $f_{n}$~is continuous, and, since $f_{n} = 0$ involves $M_{p} = 0$, that $f_{n}$~also is +ultimately of constant sign. + +Hence it is enough to establish our conclusions for functions of the +type~$M$. Let us call +\[ +\kappa_{1} + \kappa_{2} + \dots + \kappa_{h} +\] +the \emph{degree} of a term of~$M$, and let us suppose that the greatest degree +of a term of~$M$ is~$\lambda$, and that there are $\mu$~terms of degree~$\lambda$, and that +the term printed in the expression of~$M$ above is one of them. + +In the first place it is obvious, from the form of~$M$ and the fact +that $ey$~and~$ly$ are ultimately continuous when $y$~is ultimately continuous +and monotonic, that $M$~is ultimately continuous. Again, if +$M$~vanishes for values of~$x$ surpassing all limit, the same is true of +\[ +M/(\rho_{n-1} e\sigma_{n-1}), +\] +and therefore, by Rolle's theorem,\footnote + {If a function possesses a derivative for all values of its argument, the + derivative must have at least one root between any two roots of the function + itself.} +of the derivative of the latter +function. But the reader will easily verify that when we differentiate, +and arrange the terms of the derivative in the same manner as those +of~$M$, we obtain a function of the same form as~$M$ but containing at +most $\mu - 1$~terms of order~$\lambda$. And by repeating this process we clearly +arrive ultimately at a function of the form +\[ +N = \sum \rho_{n-1} e\sigma_{n-1}, +\] +%% -----File: 028.png---Folio 20------- +in which there are no factors of the form~$l\tau_{n-1}$, and which must vanish +for a sequence of values of~$x$ surpassing all limit. Hence it is +sufficient for our purpose to prove that this is impossible. + +Let the number of terms in~$N$ be~$\varpi$. Then +\[ +\frac{d}{dx} \{N/(\rho_{n-1} e\sigma_{n-1})\} +\] +must (for reasons similar to those advanced above) vanish for values +of~$x$ surpassing all limit. But when we differentiate, and arrange +the terms of the derivative in the same manner as those of~$N$, we +are left with a function of the same form as~$N$, but containing only +$\varpi - 1$~terms. And it is clear that a repetition of this process leads to +the conclusion that a function of the type +\[ +\rho_{n-1} e\sigma_{n-1} +\] +vanishes for values of~$x$ surpassing all limit, which is \textit{ex~hypothesi} +untrue. Hence the theorem is established. + +\Paragraph{3.} The proof just given, it may be observed, does not in any way +depend upon the fact that the symbols of algebraical functionality, +admitted into the definition of $L$-functions, are of an \emph{explicit} character. +We might admit such functions as +\[ +e_{2}\sqrt{ly}, +\] +where $y^{5} + y - x = 0$. But the case contemplated in the definition +seems to be the only one of any interest. + +Another interesting theorem is: \begin{Result}if $f$~is any $L$-function, we can find +an integer~$k$ such that +\[ +f \clt e_{k}x; +\] +and, if $f \cgt 1$, we can find~$k$ so that +\[ +f \cgt l_{k}x: +\] +that is to say, an $L$-function cannot increase more rapidly than any +exponential, or more slowly than any logarithm. +\end{Result} + +More precisely, an $L$-function of order~$n$ cannot satisfy $f \cgt e_{n}(x^{\Delta})$ +or $1 \clt f \clt (l_{n}x)^{\delta}$. The first part of this result is easily established; +the second appears to require a more elaborate proof. + +\Paragraph{4.} Let $f$~and~$\phi$ be any two $L$-functions which tend to infinity +with~$x$, and let $\alpha$ be any positive number. Then one of the three +relations +\[ +f \cgt \phi^{\alpha}, \qquad +f \ceqq \phi^{\alpha}, \qquad +f \clt \phi^{\alpha} +\] +must hold between $f$ and~$\phi$; and the second can hold for at most one +%% -----File: 029.png---Folio 21------- +value of~$\alpha$. If the first holds for any~$\alpha$ it holds for any smaller~$\alpha$; and +if the last holds for any~$\alpha$ it holds for any greater~$\alpha$. + +Then there are three possibilities. Either the first relation holds +for every~$\alpha$; then +\[ +f \cgt \phi^{\Delta}. +\] +Or the third holds for every~$\alpha$; then +\[ +f \clt \phi^{\delta}. +\] +Or the first holds for some values of~$\alpha$ and the third for others; and +then there is a value a of~$\alpha$ which divides the two classes of values of~$\alpha$, +and we may write +\[ +f = \phi^{\alpha} f_{1}, +\] +where $\phi^{-\delta} \clt f_{1} \clt \phi^{\delta}$. We shall find this result very useful in the +sequel. + +\Chapter[Logarithmico-Exponential Scales.] +{IV.}{Special Problems Connected with Logarithmico-Exponential Scales.} + +\begin{Remark} +\Paragraph{1. The functions $e_{r}(l_{s}x)^{\mu}$.} We have agreed to express the fact that, +however large be~$\alpha$ and however small be~$\beta$, $x^{\alpha}$~has an increase less than that +of~$e^{x^{\beta}}$, by +\[ +\Tag{(1)} +x^{\Delta} \clt e^{x^{\delta}}.\footnote + {Such a relation as + \[ + x^{\Delta_{1}} (lx)^{\Delta_{2}} \clt e^{\delta_{1} x^{\delta_{2}} (lx)^{-\Delta_{3}}} + \] + might at first sight appear to afford more information than~\Eq{(1)}: but + \[ + x^{\Delta_{1}} (lx)^{\Delta_{2}} \clt x^{\Delta_{1}'}, \qquad + \delta_{1} x^{\delta_{2}} (lx)^{-\Delta_{3}} \cgt x^{\delta_{2}'}, + \] + where $\Delta_{1}'$,~$\delta_{2}'$ are any positive numbers greater than~$\Delta_{1}$ and less than~$\delta_{2}$ respectively. + Hence our relation really expresses no more than~\Eq{(1)}.} +\] + +Let us endeavour to find a function~$f$ such that +\[ +x^{\Delta} \clt f \clt e^{x^{\delta}}. +\Tag{(2)} +\] + +If $\phi_{1} \cgt \phi_{2}$, $e^{\phi_{1}} \cgt e^{\phi_{2}}$ (\Ref{ii.}{§\;8}). Thus \Eq{(2)}~will certainly be satisfied if +\[ +\log x \clt \log f \clt x^{\delta}. +\] +Hence a solution of our problem is given by +\[ +f = e^{(\log x)^{1+\delta}}. +\] +%% -----File: 030.png---Folio 22------- + +Similarly we can prove that +\[ +f = e^{(\log x)^{1-\delta}} +\] +satisfies +\[ +(\log x)^{\Delta} \clt f \clt x^{\delta}. +\] + +It will be convenient to write +\[ +e_{0}x \eqq l_{0}x \eqq x, +\] +and then we have the relations +\[ +e_{0}(l_{1}x)^{\gamma} + \clt e_{1}(l_{1}x)^{1-\delta} + \clt e_{0}(l_{0}x)^{\gamma} + \clt e_{1}(l_{1}x)^{1+\delta} + \clt e_{1}(l_{0}x)^{\gamma}, +\Tag{(3)} +\] +where $\gamma$~denotes \emph{any} positive number.\footnote + {Here $\delta$, as usual, denotes `any positive number however small.' Of course, in + using the index~$1 - \delta$, it is tacitly implied that $\delta < 1$.} + +Let us now consider the functions +\[ +f = e_{r}(l_{s}x)^{\mu}, \qquad +f' = e_{r'}(l_{s'}x)^{\DPtypo{\mu}{\mu'}}, +\] +where $\mu$,~$\mu'$ are positive and not equal to~$1$. If $r = r'$, $f \cgt f'$ or $f \clt f'$ according +as $s < s'$ or $s > s'$. If $s = s'$, the same relations hold according as $r > r'$ or $r < r'$. +If $r = r'$ and $s = s'$, then $f \cgt f'$ or $f \clt f'$ according as $\mu > \mu'$ or $\mu < \mu'$. Leaving +these cases aside, suppose $s > s'$, $s - s' = \sigma > 0$. Putting $l_{s'}x = y$, we obtain +\[ +f = e_{r}(l_{\sigma}y)^{\mu}, \qquad +f' = e_{r'}y^{\mu'}. +\] +If $r < r'$ it is clear that $f \clt \phi$. If $r > r'$, let $r - r' = \rho$; then +\[ +l_{r}f = (l_{\sigma}y)^{\mu}, \qquad +l_{r}f' = l_{\rho}y^{\mu'} \ceqq l_{\rho}y: +\] +if $\rho > 1$ the symbol~$\ceqq$ may be replaced by~$\sim$. If $\sigma > \rho$, $l_{r}f \clt l_{r}f'$ and so +$f \clt f'$. If $\sigma < \rho$, $f \cgt f'$. If $\sigma = \rho$, $f \cgt f'$ or $f \clt f'$ according as $\mu > 1$ or +$\mu < 1$. Thus +\[ +f \cgt f' \quad (r - s > r' - s'), \qquad +f \clt f' \quad (r - s < r' - s'), +\] +while if $r - s = r' - s'$, $f \cgt f'$ or $f \clt f'$ according as $\mu > 1$ or $\mu < 1$, $\mu$~being the +exponent of the logarithm of higher order which occurs in $f$~or~$f'$. + +From this it follows that +\begin{gather*} +\dots e_{1}(l_{2}x)^{1-\delta} \clt e_{0}(l_{1}x)^{\gamma} \eqq (lx)^{\gamma} \clt e_{1}(l_{2}x)^{1+\delta} \clt e_{2}(l_{3}x)^{1+\delta} \clt \dots\\ +\dots \clt e_{2}(l_{2}x)^{1-\delta} \clt e_{1}(l_{1}x)^{1-\delta} \clt e_{0}(l_{0}x)^{\gamma} \eqq x^{\gamma} \clt e_{1}(l_{1}x)^{1+\delta} \clt \dots\\ +\dots \clt e_{3}(l_{2}x)^{1-\delta} \clt e_{2}(l_{1}x)^{1-\delta} \clt e_{1}(l_{0}x)^{\gamma} \eqq ex^{\gamma} \clt e_{2}(l_{1}x)^{1+\delta} \clt \dots\DPtypo{}{.} +\end{gather*} + +These relations enable us to interpolate to any extent among what we may +call the fundamental logarithmico-exponential orders of infinity, viz.\ $(l_{k}x)^{\gamma}$, +$x^{\gamma}$, $e_{k}x^{\gamma}$. Thus +\[ +e^{(lx)^{1+\delta}}, \quad +e^{e^{(llx)^{1+\delta}}},\ \dots, +\] +and +\[ +e^{e^{(lx)^{1-\delta}}}, \quad +e^{e^{e^{(llx)^{1-\delta}}}},\ \dots, +\] +are two scales, the first rising from above~$x^{\gamma}$, the second falling from below~$ex^{\gamma}$, +and never overlapping. + +These scales, and the analogous scales which can be interpolated between +other pairs of the fundamental logarithmico-exponential orders, possess +%% -----File: 031.png---Folio 23------- +another interesting property. The two scales written above \begin{Result}cover up \emph{(to put +it roughly)} the whole interval between $x^{\gamma}$ and~$ex^{\gamma}$, so far as $L$-functions \(\Ref{iii.}{§\;2}\) +are concerned\end{Result}: that is to say, it is impossible that an $L$-function~$f$ should +satisfy +\begin{alignat*}{2} +f &\cgt e_{r}(l_{r}x)^{1+\delta}, &&\RTag{(\emph{every} $r$),}\\ +f &\clt e_{r+1}(l_{r}x)^{1-\delta},&&\RTag{(\emph{every} $r$);} +\end{alignat*} +and the corresponding pairs of scales lying between $(l_{k+1}x)^{\gamma}$ and~$(l_{k}x)^{\gamma}$, or +between $e_{k}x^{\gamma}$ and~$e_{k+1}x^{\gamma}$, possess a similar property. This property is +analogous to that possessed (\Ref{iii.}{§\;3}) by the scales $(l_{r}x)$,~$(e_{r}x)$; viz.\ that no +$L$-function~$f$ can satisfy $f \cgt e_{r}x$, or $1 \clt f \clt l_{r}x$, for all values of~$r$. A little +consideration is all that is needed to render this theorem plausible: to +attempt to carry out the details of a formal proof would occupy more space +than we can afford. + +\Paragraph{2.} \begin{Example}\Item{(i)} Compare the rates of increase of +\[ +f = (lx)^{(lx)^{\mu}}, \qquad +\phi = x^{(lx)^{-\nu}}. +\] +\end{Example} + +These functions are the same as $e\{(lx)^{\mu}\, llx\}$,~$e\{(lx)^{1-\nu}\}$. If $\mu + \nu \geqq 1$, $f \cgt \phi$; +if $\mu + \nu < 1$, $f \clt \phi$. + +\begin{Example}\Item{(ii)} Compare the rates of increase of +\[ +f = x^{a}(lx)^{b}, \qquad +\phi = e^{A(lx)^{\alpha}(llx)^{\beta}}, \qquad +(a,\ A,\ \alpha > 0). +\] +\end{Example} +Here $f = e(a\, lx + b\, llx)$. If $\alpha < 1$, then $f \cgt \phi$; if $\alpha > 1$, then $f \clt \phi$. If $\alpha = 1$, +$\beta < 0$, then $f \cgt \phi$; if $\alpha = 1$, $\beta > 0$, then $f \clt \phi$. If $\alpha = 1$, $\beta = 0$, $a > A$, then +$f \cgt \phi$; if $\alpha = 1$, $\beta = 0$, $a < A$, then $f \clt \phi$. If $\alpha = 1$, $\beta = 0$, $a = A$, then $f \cgt \phi$ +if $b > 0$ and $f \clt \phi$ if $b < 0$. Finally if $\alpha = 1$, $\beta = 0$, $a = A$, $b = 0$ the two functions +are identical. + +\begin{Example}\Item{(iii)} Compare the increase of $f = x^{\phi/(1+\phi)}$, where $\phi$~is a function of~$x$ such +that $\phi \cgt 1$, with that of~$x^{\gamma}$. +\end{Example} + +It is clear that $f \cleq x$, but $f \cgt x^{\gamma}$ for any value of~$\gamma$ less than unity. For, +if $x$~is large enough, $\phi > n$, where $n$~is any positive integer, and so +\[ +f > x^{n/(1+n)}. +\] +Again $f = xe^{-lx/(1+\phi)}$, and so, if $\phi \clt lx$, $f \clt x$: but if $\phi \ceq lx$, $f \ceq x$; while if +$\phi \cgt lx$, $f \sim x$. + +\Paragraph{3. Successive approximations to a logarithmico-exponential function.} +Consider such a function as +\[ +f = \sqrt{x}(lx)^{2} e^{\sqrt{lx}(l_{2}x)^{2}e^{\sqrt{l_{2}x}(l_{3}x)^{2}}}. +\] +If we omit one or more of the parts of the expression of~$f$ we obtain another +function whose increase differs more or less widely from that of~$f$. The +question arises as to which parts are of the greatest and which of the least +importance; \ie\ as to which are the parts whose omission affects the increase +of~$f$ most or least fundamentally. + +Taking logarithms we find +\[ +\lf = \tfrac{1}{2}lx + \sqrt{lx}(l_{2}x)^{2} e^{\sqrt{l_{2}x}(l_{3}x)^{2}} + 2l_{2}x, +\Tag{(a)} +\] +%% -----File: 032.png---Folio 24------- +the three terms being arranged in order of importance. Again +\[ +l_{2}f = l_{2}x - l2 + \epsilon, \qquad +l_{3}f = l_{3}x + \epsilon, +\] +where (\Ref{i.}{§\;5}) in each of the last equations $\epsilon$~denotes a function (not the +same function) which tends to zero as $x \to \infty$. If we neglect this term in +each of them in turn we deduce the approximations +\[ +\ITag{(1)} f = x,\qquad +\ITag{(2)} f = \sqrt{x}. +\] + +By neglecting the last term in the equation~\Eq{(a)} we obtain the much closer +approximation +\[ +\ITag{(6)} f = \sqrt{x} e^{\sqrt{lx} (l_{2}x)^{2} e^{\sqrt{l_{2}x} (l_{3}x)^{2}}}. +\] + +In order to obtain a more complete series of approximations to~$f$ we must +replace the equation~\Eq{(a)} by a series of approximate equations. Now if +\[ +\phi = \sqrt{lx}(l_{2}x)^{2} e^{\sqrt{l_{2}x} (l_{3}x)^{2}} +\] +we have +\begin{gather*} +l\phi = \tfrac{1}{2}l_{2}x + \sqrt{l_{2}x} (l_{3}x)^{2} + 2l_{3}x,\\ +l_{2}\phi = l_{3}x - l2 + \epsilon, \qquad +l_{3}\phi = l_{4}x + \epsilon. +\end{gather*} +Hence we obtain (0)~$\phi = lx$, (3)~$\phi = \sqrt{lx}$, and (5)~$\phi = \sqrt{lx} e^{\sqrt{l_{2}x}(l_{3}x)^{2}}$ as +approximations to the increase of~$\phi$: of these, however, the first is valueless, +inasmuch as it would make~$\phi$ preponderate over the first term on the right +hand side of~\Eq{(a)}. + +A similar argument, applied to the function $e^{\sqrt{l_{2}x}(l_{3}x)^{2}}$, leads us to interpolate +(4)~$\phi = \sqrt{lx} e^{\sqrt{l_{2}x}}$ between (3)~and~(5). We can now, by adopting +a series of approximate forms of the equation~\Eq{(a)}, deduce a complete system +of closer and closer approximations to the increase of~$f$, viz.\ +\begin{gather*} +\ITag{(1)} x,\qquad +\ITag{(2)} \sqrt{x},\qquad +\ITag{(3)} \sqrt{x} e^{\sqrt{lx}},\qquad +\ITag{(4)} \sqrt{x} e^{\sqrt{lx} e^{\sqrt{l_{2}x}}},\\ +\ITag{(5)} \sqrt{x} e^{\sqrt{lx} e^{\sqrt{l_{2}x}(l_{3}x)^{2}}},\qquad +\ITag{(6)} \sqrt{x} e^{\sqrt{lx}(l_{2}x)^{2} e^{\sqrt{l_{2}x}(l_{3}x)^{2}}}. +\end{gather*} +This order corresponds exactly to the order of importance of the various parts +of the expression of~$f$. + +\Paragraph{4. Legitimate and illegitimate forms of approximation to a logarithmico-exponential +function.} In applications of this theory, such as +occur, for instance, in the theory of integral functions, we are continually +meeting such equations as +\[ +f = (1 + \epsilon)e^{x^{\alpha}}, \qquad +f = e^{(1+\epsilon)x^{\alpha}}, \qquad +f = e^{x^{\alpha+\epsilon}}, \qquad (\alpha > 0). +\Tag{(1)} +\] + +It is important to have clear ideas as to the degree of accuracy of such +representations of~$f$. The simplest method is to take logarithms repeatedly, +as in §\;3~above. + +In the first example the term~$\epsilon$ does not affect the increase of~$f$: we have +$f \sim ex^{\alpha}$. This is not true in the second; but $\lf \sim x^{\alpha}$, so that the term~$\epsilon$ does +not affect the increase of~$\lf$; while in the third this is not true, though $\llf \sim \alpha$. +Of the three formulae the first gives the most, and the last the least, information +as to the increase of~$f$ (see also \Ref{vii.}{§\;3}). +%% -----File: 033.png---Folio 25------- + +Such a formula as +\[ +f = xe^{(1+\epsilon)x^{\alpha}} +\Tag{(2)} +\] +would not be a legitimate form of approximation at all. For the factor~$e(\epsilon x^{\alpha})$ +which is not completely specified may well be far more important than the +explicitly expressed factor~$x$: we might for example have $\epsilon = x^{-\beta}$, where +$0 < \beta < \alpha$, in which case $e(\epsilon x^{\alpha})$ is more important than any power of~$x$. Thus +\Eq{(2)}~does not really convey more information than the second equation~\Eq{(1)}, +and to use it would involve a logical error similar to that involved in saying +that the sun's distance is $92,713,600$~miles, with a probable error of some +$100,000$~miles. + +\Paragraph{5. Attempts to represent orders of infinity by symbols.} It is +natural to try to devise some simple method of representing orders of +infinity by symbols which can be manipulated according to laws resembling +as far as possible those of ordinary algebra. Thus Thomae\footnote + {\textit{Elementare Theorie der analytischen Funktionen}, S.~112.} +has proposed +to represent the order of infinity of $f = x^{\alpha}(lx)^{\alpha_{1}}(l_{2}x)^{\alpha_{2}} \dots$ by +\[ +Of = \alpha + \alpha_{1}l_{1} + \alpha_{2}l_{2} + \dots,\footnote + {The reader will not confuse this use of the symbol~$O$ (which does not extend + beyond this paragraph) with that explained in \Ref{i.}{§\;5}.} +\] +where the symbols $l_{1}$,~$l_{2}$,~\dots\ are to be regarded as new units. It is clear that +these units cannot, in relation to one another, obey the Axiom of Archimedes:\footnote + {`If $x > y > 0$, we can find an integer~$n$ such that $ny > x$.'} +however great~$n$, $nl_{2}$~cannot be as great as~$l_{1}$, nor $nl_{1}$~as great as~$1$. + +The consideration of a few simple cases is enough to show that any such +notation, if it is to be of any use, must obey the following laws: +\begin{alignat*}{2} + &\Item{(i)} && \text{if} f \cgeq \phi, \qquad O(f + \phi) = Of;\\ + &\Item{(ii)} &&\quad O(f\phi) = Of + O\phi;\\ + &\Item{(iii)}\quad&&\quad O\{f(\phi)\} = Of × O\phi. +\end{alignat*} + +And Pincherle\footnote + {\lc\ (see \PageRef{p.}{13} above).} +has pointed out that these laws are in any case inconsistent +with the maintenance of the laws of algebra in their entirety. +Thus if +\[ +Ox = 1, \qquad +O\, lx = \lambda, +\] +we have, by~(iii), $O\, llx = \lambda^{2}$, and by (iii)~and~(ii), +\[ +O\, l(x\, lx) = \lambda(1 + \lambda) = \lambda + \lambda^{2}; +\] +and on the other hand, by~(i), +\[ +O\, l(x\, lx) = O(lx + llx) = \lambda. +\] + +Pincherle has suggested another system of notation; but the best yet +formulated is Borel's.\footnote + {\textit{Leçons sur les séries à termes positifs}, pp.~35 \textit{et~seq.}; for further information + see his recently published \textit{Leçons sur la théorie de la croissance}, pp.~14 \textit{et~seq.}} +Borel preserves the three laws (i),~(ii),~(iii), the +%% -----File: 034.png---Folio 26------- +commutative law of addition, and the associative law of multiplication. But +multiplication is no longer commutative, and only distributive on one side.\footnote + {$(a + b)c = ac + bc$, but in general $a(b + c) \neq ab + ac$.} +He would denote the orders of +\begin{align*} +e^{x}x^{n},&& +x^{n}(lx)^{p},&& +e^{2x},&& +e^{x^{2}},&& +e^{e^{x}},&& +e^{\sqrt{lx}},&& +\tfrac{1}{2} x, \\ +\intertext{by} +\omega + n,&& +n + \frac{p}{\omega},&& +2 · \omega,&& +\omega · 2,&& +\omega^{2},&& +\omega · \frac{1}{2} · \frac{1}{\omega},&& +\frac{1}{\omega} · \frac{1}{2} · \omega. +\end{align*} +But little application, however, has yet been found for any such system of +notation; and the whole matter appears to be rather of the nature of +a mathematical curiosity. +\end{Remark} + +\Chapter[Logarithmico-Exponential Scales.] +{V.}{Functions Which do not Conform to any Logarithmico-Exponential Scale.} + +\Paragraph{1.} \First{We} saw in \Ref{i.}{(§\;2)} that, given two increasing functions $\phi$~and~$\psi$ +($\phi \cgt \psi$), we can always construct an increasing function~$f$ which is, for +an infinity of values of~$x$ increasing beyond all limit, of the order of~$\phi$, +and for another infinity of values of~$x$ of the order of~$\psi$. The actual +construction of such functions by means of explicit formulae we left till +later. We shall now consider the matter more in detail, with special +reference to the case in which $\phi$~and~$\psi$ are $L$-functions. + +We shall say that $f$~is an \emph{irregularly increasing} function (\textit{fonction +à croissance irrégulière}) if we can find two $L$-functions $\phi$~and~$\psi$ ($\phi \cgt \psi$) +such that +\[ +f \geq \phi \quad (x = x_{1},\ x_{2},\ \dots), \qquad +f \leq \psi \quad (x = x_{1}',\ x_{2}',\ \dots), +\] +$x_{1}$,~$x_{2}$,~\dots\ and $x_{1}'$,~$x_{2}'$,~\dots\ being any two indefinitely increasing sequences +of values of~$x$. We shall also say that `the increase of~$f$ is irregular' +and that `the logarithmico-exponential scales are \emph{inapplicable} to~$f$.' + +\begin{Remark} +The phrase `\textit{fonction à croissance irrégulière}' has been defined by various +writers in various senses. Borel\footnote + {\textit{Leçons sur les fonctions entières}, p.~107.} +originally defined $f$ to be \textit{à croissance régulière} if +\[ +e^{x^{\alpha-\delta}} < f < e^{x^{\alpha+\delta}}, \RTag{$(x > x_{0})$,} +\] +or in other words if $\llf \sim \alpha lx$ or $\llf \ceqq lx$. + +This definition was of course designed to meet the particular needs of the +%% -----File: 035.png---Folio 27------- +theory of integral functions: and has been made more precise by Boutroux +and Lindelöf,\footnote + {Boutroux, \textit{Acta Mathematica}, t.~28, p.~97; Lindelöf, \textit{Acta Societatis + Fennicae}, t.~31, p.~1. See also Blumenthal, \textit{Principes de la théorie des fonctions + entières d'ordre infini}.} +who use inequalities of the form +\[ +e^{x^{\alpha}(lx)^{\alpha_{1}} \dots (l_{k}x)^{\alpha_{k}-\delta}} < f < +e^{x^{\alpha}(lx)^{\alpha_{1}} \dots (l_{k}x)^{\alpha_{k}+\delta}}. +\] + +All functions which are not \textit{à croissance régulière} for these writers are +included in our class of irregularly increasing functions. +\end{Remark} + +\Paragraph{2.} The logarithmico-exponential scales may fail to give a complete +account of the increase of a function in two different ways. The +function may be of irregular increase, as explained above, and the +scales \emph{inapplicable}: on the other hand they may be, not inapplicable, +but \emph{insufficient} (\textit{en~défaut}). That is to say, although the increase of +the function does not oscillate from that of one $L$-function to that of +another, there may be no $L$-function capable of measuring it. That +such functions exist follows at once from the general theorems of~\Ref{ii}{}. +Thus we can define a function which tends to infinity more rapidly +than any~$e_{r}x$, or more slowly than any~$l_{r}x$: and the increase of such a +function is more rapid or slower than that of any $L$-function (\Ref{iii.}{§\;2}). +Or again, we can (\Ref{ii.}{§\;6}) define a function whose increase is greater +than that of~$e_{r}(l_{r}x)^{1+\delta}$ (any~$r$) and less than that of~$e_{r+1}(l_{r}x)^{1-\delta}$ (any~$r$); +and the increase of such a function (\Ref{iv.}{§\;1}) cannot be equal to that of +any $L$-function. + +We shall now discuss some actual examples of functions for which +the logarithmico-exponential scales are inapplicable or insufficient. + +\Paragraph{3. Irregularly increasing functions.} Functions whose increase +is irregular may be constructed in a variety of ways. + +\begin{Remark} +\Item{(i)} Pringsheim\footnote + {See \textit{Math.\ Annalen}, Bd.~35, S.~347 \textit{et~seq.}\ and \textit{Münchener Sitzungsberichte}, Bd.~26, + S.~605 \textit{et~seq.}} +has used, in connection with the theory of the convergence +of series, functions of an integral variable~$n$ whose increase is +irregular. A simple example of such a function is +\[ +f(n) = 10^{[(\log_{10} n)^{1/\tau}]^{\tau}}, \RTag{$(\tau > 1)$,} +\] +where $[x]$~denotes the integral part of~$x$. It is easily proved, for instance, +when $\tau = 2$, that the increase of~$f(n)$ varies between that of~$n$ and that of +$n · 10^{1-2\sqrt{\log_{10}n}}$. We shall not do more than mention functions of this type. +They are defined, most naturally, as functions of an integral variable~$n$: if we +extend the definition to the continuous variable, the resulting function is +discontinuous. The definition can of course be modified so as to give a +%% -----File: 036.png---Folio 28------- +continuous function of~$x$ with substantially the same properties; but it is +not easy to effect this by a simple, natural, and explicit formula. + +\Item{(ii)} A more natural type of function is given by +\[ +f = \phi \cos^{2} \theta + \psi \sin^{2} \theta, +\] +where $\phi$,~$\psi$,~$\theta$ are increasing $L$-functions. We have to consider what +conditions $\phi$,~$\psi$,~$\theta$ must satisfy in order that $f$~may increase steadily with~$x$. +That its increase oscillates between that of~$\phi$ and that of~$\psi$ is obvious. + +Differentiating, +\[ +f' = \phi' \cos^{2} \theta + \psi' \sin^{2} \theta + 2(\psi - \phi)\theta' \cos \theta \sin \theta. +\] +Suppose $\phi \cgt \psi$: and let us assume that (as will be proved in the next +chapter) relations between $L$-functions involving the symbols $\cgt$,~etc.\ may be +differentiated and integrated. The condition that $f'$~should always be +positive is $\phi'\psi' \cgt (\phi - \psi)^{2}\theta'^{2}$ or $\phi'\psi' \cgt \phi^{2}\theta'^{2}$. \textit{A~fortiori}, since $\phi' \cgt \psi'$, we +must have $\phi' \cgt \phi\theta'$, or $\log\phi \cgt \theta$. Thus $f$~is certainly monotonic if +\[ +\phi \cgt \psi, \qquad +\log\phi \cgt \theta, \qquad +\psi' \cgt \phi^{2}\theta'^{2}/\phi'. +\] +If, \eg, $\theta = x$, we require $\log\phi \cgt x$, which is satisfied, for example, if +$\phi = x^{\alpha} e^{x^{\rho}}$ ($\rho > 1$). It is convenient to write $a + \rho - 1$ for~$\alpha$. Then, since +$\phi' \sim \rho x^{\alpha+\rho-1} e^{x^{\rho}}$, we must have $\psi' \cgt x^{a} e^{x^{\rho}}$; and so +\[ +\psi \cgt \int^{x} t^{a} e^{t^{\rho}}\, dt + = \frac{1}{\rho} \int^{x} t^{a-\rho+1} \frac{d}{dt}\, (e^{t^{\rho}})\, dt + \sim \frac{1}{\rho} x^{a-\rho+1} e^{x^{\rho}}, +\] +as is easily seen on integrating by parts. Thus we may take $\psi = x^{\beta} e^{x^{\rho}}$, +where $\alpha - 2\rho + 2 < \beta < \alpha$. Changing our notation a little we see that +\[ +f = (x^{\gamma+\delta} \cos^{2} x + x^{\gamma-\delta} \sin^{2} x) e^{x^{\rho}} +\] +is monotonic if $0 < \delta < \rho-1$; and the increase of~$f$ obviously oscillates +between that of~$x^{\gamma+\delta} e^{x^{\rho}}$ and that of~$x^{\gamma-\delta} e^{x^{\rho}}$. Similarly it may be shown +that +\[ +f = (e^{\mu x} \cos^{2} x + e^{\nu x} \sin^{2} x) e^{e^{x}} +\] +is monotonic if $\nu < \mu < \nu + 2$;\footnote + {Cf.\ \textit{Messenger of Mathematics}, vol.~31, p.~1.} +and again the increase of~$f$ is irregular. +\end{Remark} + +\Paragraph{4. Irregularly increasing functions (\continued).} We shall +now consider two more general and more important methods for the +construction of irregularly increasing functions. + +\Item{(iii)} Borel\footnote + {See Borel, \textit{Leçons sur les fonctions entières}, pp.~120~\textit{et~seq.}; \textit{Leçons sur les + séries à termes positifs}, pp.~32~\textit{et~seq}. Borel considers the cases only in which + $\psi = e^{x}$, $\phi = e^{x^{2}}$ or~$e^{e^{x}}$; but his method is obviously of general application. The + proof here given is however more general and much simpler.} +has shown how, by means of power series, we may +define functions which increase steadily with~$x$, while their increase +oscillates to an arbitrary extent. +%% -----File: 037.png---Folio 29------- + +Let +\[ +\phi(x) = \sum a_{n}x^{n}, \qquad +\psi(x) = \sum b_{n}x^{n} +\] +be two integral functions of~$x$ with positive coefficients; suppose also +$\phi \cgt \psi$. The increase of $\phi$~and~$\psi$ may be as large as we like (\Ref{ii.}{§\;4}); +but in each case it must be greater than that of any power of~$x$. + +Then we can define a function +\[ +f(x) = \sum c_{n}x^{n}, +\] +where every~$c_{n}$ is equal either to~$a_{n}$ or to~$b_{n}$, in such a way that, for an +infinity of values~$x_{\nu}$ whose limit is infinity, we have $f \sim \phi$, and for a +similar infinity of values~$x_{\nu}'$ we have $f \sim \psi$.\footnote + {By `$f \sim \phi$ for an infinity of values~$x_{\nu}$' we mean of course that $f/\phi \to 1$ as $x \to \infty$ + through this particular sequence of values.} + +Let $(\eta_{\nu})$ be a sequence of decreasing positive numbers whose limit is +zero. Take a positive number~$x_{0}$ such that $\phi(x_{0}) > 1$, $\psi(x_{0}) > 1$, and a +number~$x_{1}$ greater than~$x_{0}$. When $x_{1}$~is fixed we can choose~$n_{1}$ so that +\[ +\sum_{n_{1}}^{\infty} a_{n}x_{1}^{n} < \tfrac{1}{3} \eta_{1}, \qquad +\sum_{n_{1}}^{\infty} b_{n}x_{1}^{n} < \tfrac{1}{3} \eta_{1}, +\] +and so, if $c_{n}$~is either of $a_{n}$,~$b_{n}$ (however the selection may be made for +different values of~$n$), +\[ +\sum_{n_{1}}^{\infty} c_{n}x_{1}^{n} + < \sum_{n_{1}}^{\infty} (a_{n} + b_{n})x_{1}^{n} + < \tfrac{2}{3} \eta_{1}. +\] + +For $0 \leq n < n_{1}$ we take $c_{n} = a_{n}$. Then +\[ +|f(x_{1})-\phi(x_{1})| + < \sum_{n_{1}}^{\infty} (a_{n} + c_{n})x_{1}^{n} + < \eta_{1}, +\] +and so, since $\phi(x_{1}) > 1$, +\[ +\left|\frac{f(x_{1})}{\phi(x_{1})} - 1\right| < \eta_{1}. +\Tag{(1)} +\] + +Now let $x_{2}$ be a number greater than~$x_{1}$; we can suppose $x_{2}$~chosen +so that +\[ +\biggl(\,\sum_{0}^{n_{1}-1} a_{n}x_{2}^{n}\biggr) \bigg/ \psi(x_{2}) < \tfrac{1}{5} \eta_{2}, \qquad +\biggl(\,\sum_{0}^{n_{1}-1} b_{n}x_{2}^{n}\biggr) \bigg/ \psi(x_{2}) < \tfrac{1}{5} \eta_{2}. +\] +When $x_{2}$~is fixed we can choose~$n_{2}$ ($n_{2} > n_{1}$) so that +\[ +\sum_{n_{2}}^{\infty} a_{n}x_{2}^{n} < \tfrac{1}{5} \eta_{2}, \qquad +\sum_{n_{2}}^{\infty} b_{n}x_{2}^{n} < \tfrac{1}{5} \eta_{2}. +\] + +For $n_{1} \leqq n < n_{2}$ we take $c_{n} = b_{n}$. And, however $c_{n}$~be chosen for +$n \geqq n_{2}$, we have +\[ +\sum_{n_{2}}^{\infty} c_{n}x_{2}^{n} + < \sum_{n_{2}}^{\infty} (a_{n} + b_{n})x_{2}^{n} + < \tfrac{2}{5} \eta_{2}. +\] +%% -----File: 038.png---Folio 30------- +Also +\begin{align*} +%[** TN: Not aligned in the original] +|f(x_{2}) - \psi(x_{2})| + &< \sum_{0}^{n_{1}-1} a_{n}x_{2}^{n} + + \sum_{0}^{n_{1}-1} b_{n}x_{2}^{n} + + \sum_{n_{2}}^{\infty} c_{n}x_{2}^{n} + + \sum_{n_{2}}^{\infty} b_{n}x_{2}^{n} \\ + &< \tfrac{2}{5} \eta_{2} \psi(x_{2}) + + \tfrac{3}{5} \eta_{2} + < \eta_{2}\psi(x_{2}), +\end{align*} +and so +\[ +\left|\frac{f(x_{2})}{\psi(x_{2})} - 1\DPtypo{}{\right|} < \eta_{2}. +\Tag{(2)} +\] + +It is plain that, by a repetition of this process, we can find a +sequence $x_{1}$,~$x_{2}$, $x_{3}$,~\dots\ whose limit is infinity, so that +\[ +% [** TN: Semantic \RTags, but using \RTag entails ad hoc spacing] +\left|\frac{f(x_{3})}{\phi(x_{3})} - 1\right| < \eta_{3} +\quad (3),\qquad +% +\left|\frac{f(x_{4})}{\psi(x_{4})} - 1\right| < \eta_{4} +\quad(4),\qquad +\dots; +\] +and our conclusion is thus established. Incidentally we may remark +that not only $f$~itself, but all its derivatives also, are increasing and +continuous. + +It is clear that, if we were given any number of integral functions +$\phi_{1}$,~$\phi_{2}$, \dots,~$\phi_{k}$, with positive coefficients, we could define~$f$ so that +$f/\phi_{s} \to 1$, as $x \to \infty$ through a suitably chosen sequence of values, for +each of the functions~$\phi_{s}$. + +\begin{Remark} +\Item{(iv)} \textbf{Power series with gaps.} There is another method of constructing +irregularly increasing functions by means of power series which, though less +general theoretically than that explained above, is in some ways more +interesting, inasmuch as the functions to which it leads us are of a far +simpler and more natural type. We shall confine ourselves here to explaining +in general terms the general principle of the method and indicating +a few simple examples.\footnote + {For fuller details see Hardy, \textit{Proc.\ Lond.\ Math.\ Soc.}, vol.~2, pp.~332~\textit{et~seq.}; + \textit{Messenger of Mathematics}, vol.~39, p.~28; Borel, \textit{Rendiconti del Circolo Matematico + di Palermo}, t.~23, p.~320; \textit{Leçons sur la théorie de la croissance}, pp.~111~\textit{et~seq.}; + Blumenthal, \textit{Principes de la théorie des fonctions entières d'ordre infini}, pp.~5~\textit{et~seq.}} + +Let +\[ +\phi(x) = \sum a_{n}x^{n} +\Tag{(1)} +\] +be an integral function with positive coefficients: suppose, to fix our ideas, +that the coefficients decrease steadily as $n$~increases. Suppose also that, for +a particular value of~$x$, +\[ +\varpi(x) = a_{\nu} x^{\nu} +\] +is the greatest term of the series. In general one term will be the greatest, +but for certain particular values of~$x$, say $\xi_{1}$,~$\xi_{2}$,~\dots, two consecutive terms +will be equal.\footnote + {We leave aside the possibility, which obviously applies only to particular + cases, of more than two terms being equal.} +%% -----File: 039.png---Folio 31------- + +As $x$~increases, the index~$\nu$ of~$\varpi(x)$ increases, and tends to~$\infty$ with~$n$: it +thus defines a function~$\nu(x)$ such that +\[ +\nu(x) = i,\quad (\xi_{i} < x < \xi_{i+1}). +\] +At the point of discontinuity~$\xi_{i}$, where $\nu(x)$~jumps from $i - 1$ to~$i$, we may +assign to it the value~$i$. When $\nu$~is thus defined for all values of~$x$, or $\varpi(x)$~defines +a function of~$x$ which tends continuously and steadily to~$\infty$ with~$x$. + +The increase of~$\phi$ is obviously at least as great as that of~$\varpi$; it may be +expected to be greater: but it is, in ordinary cases, not so very much +greater---the increase of~$\varpi$ gives a very fair approximation to that of~$\phi$. +Thus, if $\phi(x) = e^{x}$, $a_{n} = 1/n!$, and $\xi_{i} = i$. And for $i < x < i +1$ we have +\[ +e^{i} < \phi < e^{i+1}, \qquad +(1 - \epsilon_{i}) \frac{e^{i}}{\sqrt{2\pi i}} < \varpi < (1 + \epsilon_{i}) \frac{e^{i+1}}{\sqrt{2\pi i}}.\footnote + {The second pair of inequalities are an immediate consequence of Stirling's + theorem, that $i! \sim i^{i+\frac{1}{2}} e^{-i} \sqrt{2\pi}$.}% +\] +Thus $\phi \cgt \varpi$, but $\log\phi \sim \log\varpi$: the difference between the increases of $\phi$ +and~$\varpi$ is small compared with the increases themselves. + +Now let +\[ +f(x) = \sum a_{\chi(n)} x^{\chi(n)}, +\Tag{(2)} +\] +where $\chi(n) \cgt n$: and let $p(x)$ be the function related to~$f$ as $\varpi(x)$~is to~$\phi$. +The laws of increase of~$\varpi(x)$ and of~$p(x)$ may be expected to be very much +the same, for $p(x)$~is defined by a selection from \emph{some} of the terms from \emph{all} +of which $\varpi(x)$~was selected. The increase of~$f(x)$ clearly cannot be greater, +and may be expected to be less, than that of~$\phi(x)$: but it cannot be less than +that of~$p(x)$. Hence we may expect relations of the type +\[ +p \ceq \varpi \clt f \clt \phi.\footnote + {We \emph{must} have $p \cleq \varpi$, $p \cleq f$, $\varpi \cleq \phi$, $f \cleq \phi$.} +\] +Also it is clear that, the more rapidly we suppose $\chi(n)$~to increase, the lower +in the gap between $\varpi$ and~$\phi$ will $f$~sink, and that, if we suppose $\chi$~to increase +with sufficient rapidity, we may expect to find $\varpi \ceq f$, so that the increase of~$f$ +is completely dominated by that of one (variable) term. + +We then shall have +\[ +f(x) \ceq a_{N(x)}x^{N(x)}, +\] +where $N(x)$~is a function of~$x$ which assumes successively each of a series of +integral values~$N_{i}$, so that +\[ +N(x) = N_{i}, \RTag{$(x_{i} \leqq x < x_{i+1})$.\footnotemark} +\] +\footnotetext{$N_{i}$,~$x_{i}$ are, of course, not the same as $\nu_{i}$,~$\xi_{i}$ above.}% +But, as $x$~increases from $x_{i}$ to~$x_{i+1}$, the order of~$a_{N_{i}}x^{N_{i}}$, considered as a +function of~$x$, may vary considerably, since $N_{i}$, though depending on the +%[** TN: Hardy's notation for a closed interval; inconsistent, not modernizing] +interval $(x_{i}, x_{i+1})$, does not depend on the particular position of~$x$ in that +interval. And so it is clear that we are in this way likely to be led to +functions whose increase is irregular in the sense explained in~§\;1. +%% -----File: 040.png---Folio 32------- + +Suppose, for example, that $a_{n} = n^{-n}$, so that +\[ +\phi(x) = \sum \left(\frac{x}{n}\right)^{n} + \sim \sqrt{\frac{2\pi x}{e}} e^{x/e}.\footnote + {See \Ref{ii.}{§\;3}, and the references given in the footnote to \PageRef{p.}{10}. We might + have taken $\phi(x) = e^{x}$, but our choice of~$\phi(x)$ leads to the simplest examples.} +\] + +Here +\[ +\xi_{i} = i\left(1 + \frac{1}{i}\right)^{i+1} \sim ei, +\] +and it is easily shown that $\varpi(x) \ceq e^{x/e}$. + +Now let $\chi(n) = 2^{n}$, so that +\[ +f(x) = \sum \frac{x^{2^{n}}}{2^{n2^{n}}} = \sum v_{n} +\] +say. Then $v_{i-1} = v_{i}$ if $x = 2^{i+1}$, so that $x_{i} = 2^{i+1}$ and $N_{i} = 2^{i}$ for +\[ +2^{i+1} \leqq x < 2^{i+2}. +\] +For this range of values of~$x$, $v_{i}$~is the greatest term; when $x = 2^{i+2}$, $v_{i} = v_{i+1}$. +Further, it is not difficult to show that $f(x) \ceq p(x) = v_{i}$, the behaviour of~$f(x)$ +being dominated by that of its greatest term.\footnote + {We may say roughly that \emph{in general} $f \sim p$---that is to say, $f/p \to 1$ as~$x \to \infty$ + through any sequence of values not falling inside any of certain intervals surrounding + the values~$\xi_{i}$. At a point~$\xi_{i}$, $f/p$~is nearly equal to~$2$.} + +If we put $x = 2^{i+1+\theta}$, where $0 < \theta < 1$, we find +\[ +f(x) \ceq v_{i} = 2^{(1+\theta)2^{i}} = 2^{\alpha x}, +\] +where $\alpha = (1 + \theta)2^{-1-\theta}$. This is a maximum when $1 + \theta = 1/(\log 2)$, when it +is equal to~$1/(e\log 2) = .53\dots$. Hence the increase of~$f(x)$ oscillates (roughly) +between that of~$2^{.53\dots x}$ and~$2^{\frac{1}{2}x + 1}$.\footnote + {The latter function is multiplied by~$2$, as there are two equal terms when + $\theta = 0$ or~$1$.} + +Similar considerations may be applied to the more general series +\[ +\sum \frac{x^{a^{n}}}{b^{na^{n}}}, +\] +where $a$~is an integer greater than unity. This series is derived from $\sum (x/n^{a})^{n}$, +where $\alpha = (\log b)/(\log a)$, by taking $\chi(n) = a^{n}$. Another example of an irregularly +increasing function defined in a similar manner is +\[ +f(x) = \sum \frac{x^{n^{3}}}{(n^{3})!}, +\] +the increase of which oscillates between the increases of~$e^{x}/\sqrt{x}$ and +\[ +x^{-\frac{1}{2}} e^{x-\frac{9}{8}x^{1/3}}.\footnote + {\textit{Messenger of Mathematics}, vol.~39, p.~28.} +\] +These examples are of course typical of a large class of functions. + +Before we leave this subject let us call attention to a point of considerable +%% -----File: 041.png---Folio 33------- +interest suggested by the foregoing examples. In forming the logarithmico-exponential +scales we started from the scale $x$,~$x^{2}$,~\dots\ and then formed the +function~$\sum \dfrac{x^{n}}{n!}$. If we had started, as we equally well might have done, from +the scale $x^{2}$,~$x^{4}$, $x^{8}$,~\dots\ (cf.~\Ref{ii.}{§\;1}), we should have been led to choose, as a +function transcending this scale, not~$e^{x}$ but some such function as +\[ +\sum \frac{x^{2^{n}}}{(2^{n})!}. +\] +\emph{This is one of the irregularly increasing functions of the type just considered.} +Had we proceeded thus, and completed the construction of our fundamental +scales on similar lines, our fundamental functions would for the most part +have been among those which do not conform to the logarithmico-exponential +scale, and it would have been the functions of that scale that would have +appeared as irregularly increasing functions. +\end{Remark} + +\Paragraph{5. Functions which transcend the logarithmico-exponential +scales.} We turn our attention now to functions for which +the logarithmico-exponential scales are not inapplicable but \emph{insufficient} +(§\;2). Of the existence of such functions we are already assured. +Thus a function which assumes the values $e_{1}(1)$,~$e_{2}(2)$, \dots,~$e_{\nu}(\nu)$,~\dots\ for +$x = 1$, $2$,~\dots, $\nu$,~\dots\ certainly has an increase greater than that of any +logarithmico-exponential function. No such function, however, has as +yet made its appearance naturally in analysis; it will be sufficient, +therefore, to mention two examples of such functions which transcend +the logarithmico-exponential scales in quite different manners. + +\Item{(i)} The series +\[ +\sum \frac{e_{\nu}(x)}{e_{\nu}(\nu)} +\] +has obviously, if it converges, an increase greater than that of any~$e_{\nu}(x)$. +Suppose $k - 1 \leqq x < k$. Then +\[ +\frac{e_{k}(x)}{e_{k}(k)} < 1, \qquad +\frac{e_{k+\nu}(x)}{e_{k+\nu}(k+\nu)} + < \frac{e_{k+\nu}(k)}{e_{k+\nu}(k+\nu)} + < \frac{e_{k+\nu}(k)}{e_{k+\nu}(k+1)}. +\] +But, by the Mean Value Theorem, +\[ +e_{k+\nu}(k+1) = e_{k+\nu}(k) + e_{k+\nu}(y)e_{k+\nu-1}(y) \dots e_{2}(y)e_{1}(y), +\] +where $y$~is some number between $k$~and~$k + 1$; and so +\[ +e_{k+\nu}(k+1) > e_{k+\nu}(k)e_{k+\nu-1}(k) \dots e_{1}(k). +\] +It follows that the terms of the series +\[ +\sum_{\nu=k}^{\infty} \frac{e_{\nu}(x)}{e_{\nu}(\nu)} +\] +are less than those of the series +\[ +1 + \sum_{\nu=1}^{\infty} \frac{1}{e_{1}(k)e_{2}(k) \dots e_{k+\nu-1}(k)}, +\] +%% -----File: 042.png---Folio 34------- +which is plainly convergent, and therefore that the original series is +convergent; and it is obviously only one of a large class of series +possessing similar properties. + +\begin{Remark} +(ii) Let $\phi(x)$~be an increasing function such that $\phi(0) > 0$, $\phi \cgt x$. We +can define an increasing function~$f$, which satisfies the equation +\[ +\ff(x) = \phi(x), +\Tag{(1)} +\] +as follows. + +Draw the curves $y = x$, $y = \phi(x)$ (\Fig{5}). Take $Q_{0}$~arbitrarily on~$OP_{0}$ (see +the figure); draw~$Q_{0}R_{1}$ parallel +to~$OX$ and complete the rectangle~$Q_{0}Q_{1}$. +Join $Q_{0}$,~$Q_{1}$ by any +continuous arc everywhere inclined +at an acute angle to the +axes. On this arc take any +point~$Q$; draw $QP$,~$QR$ parallel +to the axes, and complete the +rectangle~$QQ'$. As $Q$~moves +from $Q_{0}$ to~$Q_{1}$, $Q'$~moves from +$Q_{1}$ to~$Q_{2}$, say. As we constructed +$Q'$ from~$Q$, so we can +construct $Q''$ from~$Q'$: proceeding +thus we define a continuous +curve $Q_{0}Q_{1}Q_{2}Q_{3}\dots$ corresponding +to a continuous and increasing +function~$f(x)$. Then +$f(x)$~satisfies~\Eq{(1)}. For if $y = f(x)$ +is the ordinate of~$Q$, it is clear that $\ff(x)$~is the ordinate of~$Q'$, which is equal +to~$\phi(x)$, the ordinate of~$P$. +%[Illustration: Fig. 5] +\Figure[0.6\textwidth]{5}{042} + +Let us write +\[ +f(x) = f_{1}(x), \qquad +\phi(x) = f_{1}f_{1}(x) = f_{2}(x), \qquad +f\phi(x) = \phi f(x) = f_{3}(x), +\] +and so on, so that $Q_{n}$~is the point $f_{n}(0)$,~$f_{n+1}(0)$. Also let $\psi$~be the function +inverse to~$\phi$, and write~$\psi_{2}$ for~$\psi\psi$, and so on. Finally, let the equation of~$Q_{0}Q_{1}$ +be $\theta(x, y) = 0$. Then it is easy to see that the equations of~$Q_{2n}Q_{2n+1}$ +and of~$Q_{2n+1}Q_{2n+2}$ are respectively +\[ +\theta\{\psi_{n}(x), \psi_{n}(y)\} = 0, \qquad +\theta\{\psi_{n+1}(y), \psi_{n}(x)\} = 0. +\] + +Suppose for example that $\phi(x) = e^{x}$, $OQ_{0} = a < 1$, and that $Q_{0}Q_{1}$~is the +straight line $y = a + \alpha x$, where $\alpha = (1 - a)/a$. Then the equations of~$Q_{2n}Q_{2n+1}$ +and of~$Q_{2n+1}Q_{2n+2}$ are +\[ +l_{n}y = a + \alpha l_{n}x, \qquad +l_{n}x = a + \alpha l_{n+1}y, +\] +or +\[ +y = e_{n-1}\{e^{\alpha} (l_{n-1}x)^{\alpha}\}, \qquad +y = e_{n}\{e^{-a/\alpha} (l_{n-1}x)^{1/\alpha}\}. +\] +%% -----File: 043.png---Folio 35------- +For simplicity let us take $a = \frac{1}{2}$, $\alpha = 1$. Then the equations of~$Q_{2n}Q_{2n+1}$ and +of~$Q_{2n+1}Q_{2n+2}$ are respectively +\begin{alignat*}{3} +y &= e_{n-1}\{\sqrt{e}(l_{n-1}x)\} &&= e_{n-2}\{(l_{n-2}x)^{\sqrt{e}}\} &&= \lambda_{n}(x),\\ +y &= e_{n}\{(l_{n-1}x)/ \sqrt{e}\} &&= e_{n-1}\{(l_{n-2}x)^{1/\sqrt{e}}\} &&= \mu_{n}(x), +\end{alignat*} +say. Now (\Ref{iv.}{§\;1}) +\[ +x^{\gamma} \clt \lambda_{3} \clt \dots \clt \lambda_{n} \clt \dots + \clt \mu_{n} \clt \dots \clt \mu_{3} \clt e^{x^{\gamma}} +\] +and a function~$f$, such that $\lambda_{n} \clt f \clt \mu_{n}$ for all values of~$n$, transcends the +logarithmico-exponential scales. But $f$~clearly satisfies these relations, and +so its increase is incapable of exact measurement by these scales. + +It is easily verified that $\lambda_{n}\lambda_{n}x \clt e^{x}$ and $\mu_{n}\mu_{n}x \cgt e^{x}$ for all values of~$n$. +Hence it is clear \textit{a~priori} that any increasing solution of~\Eq{(1)} must satisfy +$\lambda_{n} \clt f \clt \mu_{n}$. + +This kind of `graphical' method may also be employed to define functions +whose increase, like that of the function considered under (i) above, is slower +than that of any logarithm or more rapid than that of any exponential. It +can be employed, for example, to solve the equation +\[ +\phi(2^{x}) = 2\phi(x); +\] +and it can be proved that the increase of a function such that $\phi(2^{x}) \ceq \phi(x)$ +is slower than that of any logarithm (\Ref{vii.}{§\;3}). +\end{Remark} + +\Paragraph{6. The importance of the logarithmico-exponential scales.} +As we have seen in the earlier paragraphs of this section, it is possible, +in a variety of ways, to construct functions whose increase cannot be +measured by any $L$-function. It is none the less true that no one yet +has succeeded in defining a mode of increase genuinely independent of +all logarithmico-exponential modes. Our irregularly increasing functions +oscillate, according to a logarithmico-exponential law of oscillation, +between two logarithmico-exponential functions; the functions of~§\;5 +were constructed expressly to fill certain gaps in the logarithmico-exponential +scales. No function has yet presented itself in analysis +the laws of whose increase, in so far as they can be stated at all, cannot +be stated, so to say, in logarithmico-exponential terms. + +It would be natural to expect that the arithmetical functions which +occur in the theory of the distribution of primes might give rise to +genuinely new modes of increase. But, so far as analysis has gone, the +evidence is the other way. + +\begin{Remark} +Thus if we denote by~$\varpi(x)$ the number of prime numbers less than~$x$, it is +known that +\[ +\varpi(x) \sim \frac{x}{\log x}. +\] +%% -----File: 044.png---Folio 36------- + +More precisely +\[ +\varpi(x) = \int_{2}^{x} \frac{dt}{\log t} + \rho(x) = \Li(x) + \rho(x), +\] +where $|\rho(x)| \clt x(\log x)^{-\Delta}$. The precise order of~$\rho(x)$ has not yet been +determined, but there is reason to anticipate that $\rho(x) \cleq \sqrt{x}/(\log x)$. +\end{Remark} + + +\Chapter{VI.}{Differentiation and Integration.} + +\Paragraph{1. Integration.} It is important to know when relations of the +types $f(x) \cgt \phi(x)$, etc., can be differentiated or integrated. The +results are very much what might be expected from analogy with +similar results in other branches of analysis, and may therefore be +discussed somewhat summarily. For brevity we denote +\[ +\int_{a}^{x} f(t)\, dt, \qquad +\int_{a}^{x} \phi(t)\, dt +\] +(where $a$~is a constant) by $F(x)$ and~$\Phi(x)$. And we suppose for the +moment that $f$ and~$\phi$ are positive for $x \geqq a$. + +It may be well to repeat (cf.~\Ref{i.}{§\;4}) that $f$ and~$\phi$ are always supposed +to be (at any rate for $x > x_{0}$) positive, continuous, and monotonic, unless +the contrary is stated or clearly implied. Some of our conclusions are +valid under more general conditions; but the case thus defined, and +the corresponding case in which $f$ or~$\phi$ or~both of them are negative, +are the only cases of importance. + +\begin{Lemma} +If $\Phi \cgt 1$, and $f > H\phi$ for $x > x_{0}$, then $x_{1}$~can be found +so that $F > (H - \delta)\Phi$ for $x > x_{1}$: similarly $f < h\phi$ for $x > x_{0}$ involves +$F < (h + \delta)\Phi$ for $x > x_{1}$. +\end{Lemma} + +For if $f > H\phi$ for $x > x_{0}$, we have +\[ +F = \int_{a}^{x} f\, dt + > \int_{a}^{x_{0}} f\, dt + H \int_{x_{0}}^{x} \phi\, dt + > H\Phi + \int_{a}^{x_{0}} f\, dt - H \int_{a}^{x_{0}} \phi\, dt, +\] +and if we choose $x_{1}$ so that +\[ +\left(\int_{a}^{x_{0}} f\, dt + H \int_{a}^{x_{0}} \phi\, dt\right) \bigg/ \Phi < \epsilon +\] +for $x \geq x_{1}$, as we certainly can if $\Phi \cgt 1$, the result follows. Similarly +in the other case. From this lemma we can at once deduce the +following +%% -----File: 045.png---Folio 37------- + +\begin{Theorem} +Any one of the relations +\begin{alignat*}{5} +f &\cgt \phi, \qquad& +f &\clt \phi, \qquad& +f &\ceq \phi, \qquad& +f &\ceqq \phi, \qquad& +f &\sim \phi \\ +\intertext{involves the corresponding one of the relations} +F &\cgt \Phi, \qquad& +F &\clt \Phi, \qquad& +F &\ceq \Phi, \qquad& +F &\ceqq \Phi, \qquad& +F &\sim \Phi +\end{alignat*} +if either $F \cgt 1$ or $\Phi \cgt 1$. +\end{Theorem} + +To this we may add: \begin{Result}if both $\ds\int^{\infty} f\,dt$, $\ds\int^{\infty} \phi\,dt$ are convergent, then +$f \cgt \phi$, $f \clt \phi$, $f \ceq \phi$, $f \ceqq \phi$, $f \sim \phi$ involve corresponding relations between +\[ +\bar{F} = \int_{x}^{\infty} f\,dt, \qquad +\bar{\Phi} = \int_{x}^{\infty} \phi\,dt. +\] +\end{Result} + +The proof we may leave to the reader. These results have been +stated primarily for the case in which $f$~and~$\phi$ are positive; but there +is no difficulty in extending them to the case in which either function +or both are negative. + +\Paragraph{2. Differentiation.} {\Loosen It follows from~§\;1 that $f \cgt \phi$ involves +$f' \cgt \phi'$ if $f \cgt 1$ or $f \clt 1$ and \emph{if any one of the relations expressed by +$\cgt$,~$\clt$, $\ceq$,~$\ceqq$,~$\sim$ holds between $f'$~and~$\phi'$}.} + +\begin{Remark} +In interpreting this statement regard must be paid to the conventions +laid down in \Ref{i.}{§\;4}. Thus if $f \cgt \phi \cgt 1$, $f'$~and~$\phi'$ are positive; and $f' \cgt \phi'$. +But if $f \cgt 1 \cgt \phi$, $\phi$~is a decreasing function and $\phi' < 0$. In this case +$f' \cgt -\phi'$, a relation which we have agreed to denote by $f' \cgt \phi'$. If $1 \cgt f \cgt \phi$ +both $f'$~and~$\phi'$ are negative: the relation $-f' \clt -\phi'$ would involve +\[ +-\int_{x}^{\infty} f'\,dt \clt -\int_{x}^{\infty} \phi'\,dt +\] +or $f \clt \phi$, and is therefore impossible; similarly for $-f' \ceq -\phi'$; so we must +have $-f' \cgt -\phi'$, a relation which we have agreed also to denote by $f' \cgt \phi'$. +The case in which $f \ceq 1$ is exceptional; any one of the relations $f' \cgt \phi'$,~etc.\ +may then hold. Thus if $f = 1 + e^{-x}$, $f' = 1/x$, we have $f \cgt \phi$, $f' \clt \phi'$. The fact +is that in this case $f$, regarded as the integral of~$f'$, is dominated by the +constant of integration. +\end{Remark} + +Similar results hold, of course, for the relations $f \clt \phi$,~etc., with +similar exceptions. With regard to all of them it is to be observed +that the assumption that one of the relations holds between $f'$~and~$\phi'$ +is essential. We can never \emph{infer} that one of them holds. +We cannot even infer that $f'$~or~$\phi'$ is a steadily increasing or decreasing +function at all. Thus if $f = e^{x}$, $\phi = e^{x} + \sin e^{x}$, we have $f' = e^{x}$ and +$\phi' = e^{x}(1 + \cos e^{x})$. Thus $f$~and~$\phi$ increase steadily and $f \sim \phi$, $f' \sim f$; +%% -----File: 046.png---Folio 38------- +but $\phi'$~does not tend to infinity (vanishing for an infinity of values +of~$x$). Again if +\[ +\phi = e^{x}(\sqrt{2} + \sin x) + \tfrac{1}{2} x^{2}, +\] +we have +\[ +\phi' = e^{x} (\sqrt{2} + \sin x + \cos x) + x +\] +and $\phi \ceq e^{x}$, while $\phi'$~oscillates between the orders of $e^{x}$ and~$x$. It is +possible, though less easy, to obtain examples of this character in which +$\phi'$~also is monotonic. + +\Paragraph{3. Differentiation of $L$-functions.} If $f$~and~$\phi$ are $L$-functions, +so are $f'$~and~$\phi'$, and one of the relations $f' \cgt \phi'$, $f' \ceq \phi'$, $f' \clt \phi'$ +certainly holds (\Ref{iii.}{§\;2}). Thus in this case \emph{both differentiation and +integration are always legitimate}\footnotemark---this statement, however, being +subject to certain exceptions in the cases in which $f \ceq 1$ or $\phi \ceq 1$. +\footnotetext{A tacit assumption to this effect underlies much of Du~Bois-Reymond's work.} + +In what follows we shall suppose that all the functions concerned +are $L$-functions, or at any rate resemble $L$-functions in so far that one +of the relations $f \cgt \phi$, $f \ceqq \phi$, $f \clt \phi$ is bound to hold between any pair +of functions, and that differentiation and integration are permissible.\footnote + {The results which follow are all in substance due to Du~Bois-Reymond.} + +\begin{Result}[1.] If $f$~is an increasing function, and $f' \cgt f$, then $f \cgt e^{\Delta x}$. If +$f' \clt f$, then $f \clt e^{\delta x}$. Similarly if $f$~is a decreasing function, $f' \cgt f$ and +$f' \clt f$ involve $f \clt e^{-\Delta x}$ and $f \cgt e^{-\delta x}$ respectively. If $f' \ceqq f$, then +$e^{\delta x} \clt f \clt e^{\Delta x}$ or $e^{-\Delta x} \clt f \clt e^{-\delta x}$, and we can find a number~$\mu$ such +that $f = e^{\mu x} f_{1}$, where $e^{-\delta x} \clt f_{1} \clt e^{\delta x}$. +\end{Result} + +The proofs of these assertions are almost obvious. Thus if $f$~is an +increasing function, and $f' \cgt f$, we have +\[ +f'/f \cgt 1, \qquad +\log f \cgt x, +\] +and so $\log f > \Delta x$ for $x > x_{0}$, \ie\ $f > e^{\Delta x}$, or, what is the same thing, +$f \cgt e^{\Delta x}$. The last clause of the theorem follows at once from~\Ref{iii.}{§\;4}. + + +\begin{Result}[2.] More generally, if $v$~is any increasing function, $f'/f \cgt v'/v$ +involves $f \cgt v^{\Delta}$ or $f \clt v^{-\Delta}$, according as $f$~is an increasing or a decreasing +function; and $f'/f \clt v'/v$ involves $f \clt v^{\delta}$ or $f \cgt v^{-\delta}$. And $f'/f \ceqq v'/v$ +involves $v^{\delta} \clt f \clt v^{\Delta}$ or $v^{-\Delta} \clt f \clt v^{-\delta}$; and then we can find a number~$\mu$ +such that $f = v^{\mu}f_{1}$, where $v^{-\delta} \clt f_{1} \clt v^{\delta}$. +\end{Result} + +When $f$~is an increasing function we shall call $f'/f$ the \emph{type}~$t$ of~$f$:\footnote + {Du~Bois-Reymond calls $f/f'$ the type; the notation here adopted seems slightly + more convenient.} +it being understood that $t$~may be replaced by any simpler function~$\tau$ +such that $t \ceqq \tau$. The type of a \emph{decreasing} function~$f$ we define to be +%% -----File: 047.png---Folio 39------- +the same as that of the increasing function~$1/f$. The following table +shews the types of some standard functions: +\[ +\begin{array}{lcccccrlcc} +\text{\textit{Function}} & 1 & llx & lx & x^{\alpha} & e^{x} & e^{\alpha x^{\beta}} & e_{2}x & e_{3}x & \dots \\ +\text{\textit{Type}} & 0 & \dfrac{1}{x\, lx\, llx} & \dfrac{1}{x\, lx} & \dfrac{1}{x} & 1 & x^{\beta-1} & ex & e_{2}x\,ex & \dots +\end{array} +\] + +\begin{Remark} +If $f \cgt \phi$, then $f'/f \cgeq \phi'/\phi$. By making the increase of~$f$ large enough we +can make the increase of $t = f'/f$ as large as we please. The reader will find +it instructive to write out formal proofs of these propositions, and also of +the following. + +\Item{1.} As the increase of~$f$ becomes smaller and smaller, $f'/f$~tends to zero +more and more rapidly, but, so long as $f \to \infty$ at all, we cannot have +\[ +f'/f \clt \phi, \qquad +\int^{\infty} \phi\, dx \quad \text{\emph{convergent}}. +\] +On the other hand, if the last integral is divergent we can find~$f$ so that +$f \cgt 1$, $f'/f \clt \phi$. + +\Item{2.} Although we can find~$f$ so that $f'/f$~shall have an increase larger than +that of any given function of~$x$, we cannot have +\[ +f'/f \cgt \phi(f), \qquad +\int^{\infty} \frac{dx}{x\phi(x)} \quad \text{\emph{convergent}}. +\] +On the other hand, if the last integral is divergent we can find~$f$ so that +$f'/f \cgt \phi(f)$. + +{\Loosen[Thus we cannot find a function~$f$ which tends to infinity so slowly that +$f'/f \clt 1/x^{\alpha}$ ($\alpha > 1$). But we can find~$f$ so that $f'/f \clt 1 / x\, lx\, llx$ (\eg~$f = l_{3}x$). +We cannot find~$f$ so that $f'/f \cgt f^{\alpha}$ or $f' \cgt f^{1+\alpha}$ ($\alpha > 0$). But we can find~$f$ +so that $f'/f \cgt \lf$ (\eg~$f = e_{3}x$).]} + +\Item{3.} If $f \cgt e_{k}x$ for all values of~$k$, $f'/f$~satisfies the same condition, and +\[ +f' \cgt f\, \lf\, l_{2}f \dots l_{k}f. +\] + +He will also find it profitable to formulate corresponding theorems about +functions of a positive variable~$x$ which tends to zero. +\end{Remark} + +\Paragraph{4. Successive differentiation.} Du~Bois-Reymond has given +the following general theorem, which enables us to write down the +increase of any derivative of any logarithmico-exponential function. +We write $t$ for~$f'/f$, as in the last section, and we assume that no +derivative~$f^{(n)}$ satisfies $f^{(n)} \ceqq 1$: if this should be the case the results +of the theorem, so far as the derivatives $f^{(n+1)}$,~\dots\ are concerned, cease +to be true. + +\begin{Theorem} \Item{(i)} If $t \cgt 1/x$ \(so that $f \cgt x^{\Delta}$\) then +\[ +f \ceqq f'/t \ceqq f''/t^{2} \ceqq f'''/t^{3} \dots \ceqq f^{(n)}/t^{n} \dots. +\] +%% -----File: 048.png---Folio 40------- + +\Item{(ii)} If $t \clt 1/x$ \(so that $f \clt x^{\delta}$\) then +\[ +f \ceqq f'/t \ceqq xf''/t \ceqq x^{2}f'''/t \dots \ceqq x^{n-1} f^{(n)}/t \dots. +\] + +\Item{(iii)} If $t \ceqq 1/x$ \(so that $f = x^{\mu} f_{1}$, where $x^{-\delta} \clt f_{1} \clt x^{\delta}$\), then if $\mu$~is +not integral either set of formulae is valid. But if $\mu$~is integral +\[ +f \ceqq xf' \ceqq x^{2}f'' \dots \ceqq x^{\mu}f^{(\mu)} \ceqq x^{\mu} f^{(\mu+1)}/t_{1} \ceqq x^{\mu+1}f^{(\mu+2)}/t_{1} \dots, +\] +where $t_{1}$ is the type of~$f_{1}$. +\end{Theorem} + +\Item{(i)} If $t \cgt 1/x$, $1/t \clt x$ and so $t'/t^{2} \clt 1$; hence $t'/t \clt t = f'/f$ or +\[ +ft' \clt f't. +\] + +Differentiating the relation $f' \ceqq ft$, and using the relation just +established, we obtain +\[ +f'' \ceqq f't + ft' \ceqq f't. +\] + +Thus the type of~$f'$ is the same as that of~$f$; accordingly the +argument may be repeated and the first part of the theorem follows. + +\Item{(ii)} If $t \clt 1/x$, $xf' \clt f$ and so +\[ +xf'' + f' \clt f'. +\] + +But this cannot possibly be the case unless $xf'' \ceqq f'$. Differentiating +again we infer +\[ +xf''' + 2f'' \clt f'', +\] +whence $xf''' \ceqq f''$; and so on generally.\footnote + {More precisely $xf'' \sim -f'$, $xf''' \sim -2f''$, and so on.} +Thus the second part +follows. + +\Item{(iii)} If $t \ceq 1/x$, $f = x^{\mu}f_{1}$ and $t_{1}$,~the type of~$f_{1}$, satisfies $t_{1} \clt 1/x$. +Then +\[ +f' = \mu x^{\mu-1} f_{1} + x^{\mu}f_{1}' \ceqq x^{\mu-1} f_{1}(\mu + xt_{1}) \ceqq x^{\mu-1}f_{1}; +\] +Similarly $f'' \ceqq x^{\mu-2}f_{1}$ and so on. We can proceed indefinitely in this +way unless $\mu$~is integral: in this case we find $f^{(\mu)} \ceq f_{1}$, and from this +point we proceed as in case~(ii). + +\begin{Remark} +\textit{Examples.} \Item{(i)} If $f = e^{\sqrt{x}}$, then $t = 1/\sqrt{x} \cgt 1/x$, and $f^{(n)} \ceqq e^{\sqrt{x}}/(\sqrt{x})^{n}$. +If $f = e^{(\log x)^{2}}$, then $t = (\log x)/x \cgt 1/x$, and $f^{(n)} \ceqq e^{(\log x)^{2}} (\log x)^{n}/x^{n}$. + +\Item{(ii)} If $f = (\log x)^{m}$, then $t = 1/(x\log x) \clt 1/x$, and +\[ +f^{(n)} \ceqq tx^{-(n-1)}f \ceqq (\log x)^{m-1}/x^{n}. +\] + +\Item{(iii)} If $f = x^{2}\, llx$, $t \ceqq 1/x$. Here +\[ +f' \ceqq x\, llx, \qquad +f'' \ceqq llx, \qquad +f''' \ceqq 1/x\, lx, \qquad +f'''' \ceqq 1/x^{2}\, lx,\ \dots. +\] + +\Item{(iv)} The results of the theorem, in the first two cases, can be stated +more precisely as follows: + +If $t \cgt 1/x$, then +\[ +f^{(n)} \sim (f'/f)^{n}f. +\] +%% -----File: 049.png---Folio 41------- + +If $t \clt 1/x$, then +\[ +f^{(n)} \sim (-1)^{n-1} (n - 1)!\, x^{-(n-1)}f'. +\] + +If $f$~is a positive increasing function, then if $t \cgt 1/x$ all the derivatives are +ultimately positive, while if $t \clt 1/x$ they are alternately ultimately positive +and ultimately negative. +\end{Remark} + +\Paragraph{5. Functions of an integral variable.} The theorems for +functions of an integral variable~$n$, corresponding to those of §§\;1--4, +involve sums +\[ +A_{n} = a_{1} + a_{2} + \dots + a_{n} +\] +in place of integrals, and differences +\[ +\Delta a_{n} = a_{n} - a_{n+1} +\] +instead of differential coefficients. The reader will be able to +formulate and to prove for himself the theorems which correspond +to those of~§\;1. Thus +\begin{quote}`\begin{Result}% +$a_{n} \cgt b_{n}$, $a_{n} \clt b_{n}$, $a_{n} \ceq b_{n}$, $a_{n} \ceqq b_{n}$, $a_{n} \sim b_{n}$ involve the corresponding +equations for $A_{n}$,~$B_{n}$, if one at least of $A_{n}$,~$B_{n}$ tends +to infinity with~$n$% +\end{Result}' +\end{quote} +and so on.\footnote + {This is of course the well known theorem of Cauchy and Stolz: see Bromwich, + \textit{Infinite Series}, p.~377.} +Considerations of space forbid that we should go further +into the subject here. + + +\Chapter[Developments of the Infinitärcalcül.] +{VII.}{Some Developments of Du~Bois-Reymond's +Infinitärcalcül.} + +\Paragraph{1.} \First{We} shall conclude our account of the general theory by a brief +sketch of some interesting results due in the main to Du~Bois-Reymond. +For further details we must refer to his memoirs catalogued in the +Bibliographical Appendix. + +\Section{The functions $\dfrac{f(x + a)}{f(x)}$, $\dfrac{f(ax)}{f(x)}$, etc.} + +It is often necessary to obtain approximations to such functions as +\[ +f(x + a)/f(x), +\] +where $a$~is itself a function of~$x$, which for simplicity we suppose +positive, and which may tend to infinity with~$x$. In this connection +%% -----File: 050.png---Folio 42------- +Du Bois-Reymond\footnote + {\textit{Math.\ Annalen}, Bd.~8, S.~363 \textit{et~seq.}} +has proved a whole series of theorems: it will be +sufficient for our present purpose to give a few specimens of his results. +In what follows it will be assumed throughout that all the functions +dealt with are $L$-functions, or at any rate such that any pair of them +satisfy one of the relations $f \cgt \phi$, $f \ceqq \phi$, $f \clt \phi$, and that such +relations may be differentiated or integrated. This being so we +have +\[ +\frac{f(x + a)}{f(x)} = e^{\lf(x + \alpha) - \lf(x)} = e\left\{a\frac{f'(x + \alpha)}{f(x + \alpha)}\right\}, +\] +where $0 < \alpha < a$. This expression has certainly the limit unity if +$f' \cleq f$ and $a \clt 1$. Hence +\[ +f(x + a) \sim f(x) +\Tag{(1)} +\] +if $a \clt 1$ and $e^{-\Delta x} \clt f \clt e_{\Delta x}$. If $f'/f \clt 1$, \ie\ if $e^{-\delta x} \clt f \clt e^{\delta x}$, the +relation~\Eq{(1)} holds for $a \clt f/f'$: it certainly holds, for instance, if +$a = x\{f(x)\}^{-\mu}$, where $\mu > 0$, since $x/f^{\mu} \clt f/f'$ whenever $f \cgt 1$.\footnote + {For $\ds\int^{\infty} f^{-1-\mu} f'\,dx$ is convergent, and so $f'/f^{1+\mu} \clt 1/x$.} + +If $a \ceqq f/f'$ (as \eg\ if $f = e^{\mu x}f_{1}$, where $e^{-\delta x} \clt f_{1} \clt e^{\delta x}$, and $a \ceqq 1$), +$f(x + a)/f(x)$ will tend to a limit different from unity. + +Again +\[ +\frac{f(x + a)}{f(x)} = e\left\{a\frac{f'(x)}{f(x)}\, \frac{t(x + \alpha)}{t(x)}\right\}, +\] +where $t = f'/f$. Hence +\[ +\frac{f(x + a)}{f(x)} = e\left\{u\frac{f'(x)}{f(x)}\right\} \quad (u \sim a) +\Tag{(2)} +\] +{\Loosen in all cases in which $t(x + \alpha)/t(x) \sim 1$; as for example if $a \cleq 1$, +$e^{-\delta x} \clt t \clt e^{\delta x}$, or, what is the same thing, if} +\[ +a \cleq 1, \qquad +e^{-e^{\delta x}} \clt f \clt e^{e^{\delta x}}. +\] + +The reader will find it instructive to write down conditions under +which the equation~\Eq{(2)} holds when $u \ceqq a$ is substituted for $u \sim a$, and +to consider in what circumstances either relation holds when $a \cgt 1$. + +\Paragraph{2.} The reader is also recommended to verify some of the +following results: + +\begin{Remark} +\begin{Result} +\Item{(i)} If $1 \clt a \clt x$ and $x^{-\Delta} \clt f \clt x^{\Delta}$, then $f(x + a)/f(x) \sim 1$. + +\Item{(ii)} \Squeeze{If $f \clt x$ and $a \clt 1/f'$, or if $f \ceqq x$ and $a \clt 1$, then $f(x + a) - f(x) \clt 1$}. + +\Item{(iii)} If $e^{-\delta x} \clt f \clt e^{\delta x}$ and $a \clt f'/f''$, then +\[ +f(x + a) - f(x) \sim af'(x). +\] +\end{Result} +%% -----File: 051.png---Folio 43------- + +The condition $a \clt f'/f''$ may be simplified by means of the theorem of +\Ref{vi.}{§\;4}. Thus if $t \clt 1/x$ (\ie\ if $f \clt x^{\delta}$) it is equivalent to $a \clt x$. + +\begin{Result} +\Item{(iv)} If $x^{-\delta} \clt a \clt x^{\delta}$, $(lx)^{-\Delta} \clt f \clt (lx)^{\Delta}$, then $f(ax)/f(x) \sim 1$. + +\Item{(v)} If $e^{-\Delta\sqrt{lx}} \clt f \clt e^{\Delta\sqrt{lx}}$, then +\[ +\frac{f\{xf(x)\}}{f(x)} \ceqq 1, \qquad +e\left\{\frac{x\, \lf(x)f'(x)}{f(x)}\right\} \ceqq 1; +\] +and the limits of the two functions are the same: and if $e^{-\delta\sqrt{lx}} \clt e^{\delta\sqrt{lx}}$ this +limit is unity. +\end{Result} + +Suppose, \eg\ $f \cgt 1$, and let $f(x) = \phi(lx)$; then, if $a = f(x)$, +\[ +\frac{f(ax)}{f(x)} = e^{l\phi(lx + la) - l\phi(lx)} + = e^{la\phi'(lx + la_{1})/\phi(lx + la_{1})}, +\] +where $1 < a_{1} < a$. The exponent is +\[ +l\phi(lx + la_{1}) \frac{\phi'(lx + la_{1})}{\phi(lx + la_{1})}\, \frac{l\phi(lx)}{l\phi(lx + la_{1})}. +\] + +Now $a = f(x) \clt x^{\delta}$ and therefore $la_{1} \cleq la \clt lx$, and so, by~(i), +\[ +l\phi(lx + la_{1}) \sim l\phi(lx) +\] +if $l\phi \clt x^{\Delta}$ or if $f \clt e^{(lx)^{\Delta}}$, which is certainly the case. Hence the exponent +is asymptotically equivalent to +\[ +l\phi(u) \phi'(u)/\phi(u), +\] +where $u = lx + la_{1}$. And $l\phi(\phi'/\phi) \cleq 1$ if $(l\phi)^{2} \cleq u$, \ie\ if $\phi \cleq e^{\Delta\sqrt{u}}$ or +$f \cleq e^{\Delta\sqrt{lx}}$. In this case $f(ax) \ceqq f(x)$; and it is easy to see that if +$f \cleq e^{\delta\sqrt{lx}}$ the symbol~$\ceqq$ may be replaced by~$\sim$. + +\Item{(vi)} \emph{If $f(x) = x\phi(x)$, and $e^{-\delta\sqrt{lx}} \clt \phi \clt e^{\delta\sqrt{lx}}$, then} +\[ +f_{2}(x) \eqq \ff(x) \sim x\phi^{2},\ \dots,\ f_{n} \sim x\phi^{n},\ \dots. +\] + +The reader will easily prove this by the aid of the preceding results. He +will also find it instructive to calculate the increase of~$f_{n}$ when $f = e^{\sqrt{lx}}$ and +when $f = e^{(lx)^{\alpha}}$, where $\alpha > \frac{1}{2}$. +\end{Remark} + +\Section{The accuracy of approximations.} + +\Paragraph{3.} We have already (\Ref{iv.}{§§\;3--4}) had occasion to use the notion +of an approximation to the increase of a function, and to distinguish +legitimate and illegitimate forms of approximation. Du~Bois-Reymond +has given the following more precise definitions. + +He defines $\psi(x, u, u_{1}, \dots)$ to be an `approximate form' of~$y$ if +\[ +y = \psi(x, u, u_{1}, \dots), +\] +$\psi$~being a known function, and $u$,~$u_{1}$,~\dots\ unknown functions whose +increase is, however, subject to certain limitations. It is clear that +it is really useless, however, to insert more than one unknown function~$u$ +%% -----File: 052.png---Folio 44------- +in~$\psi$. The effect of the presence of~$u$ is to define a certain stretch +within which the increase of~$y$ lies, and the presence of several~$u$'s can +effect no more. We shall therefore consider only approximate forms +of the type +\[ +y = \psi(x, u). +\Tag{(1)} +\] + +Thus +\[ +e^{x^{u}} \quad (u \sim 1), \qquad +e^{(1+u)x} \quad (u \clt 1), \qquad +x^{1+u}e^{x} \quad (u \clt 1) +\Tag{(2)} +\] +are approximate forms of $y = xe^{x}/lx$; the second clearly closer than +the first and the third than the second. + +The closeness of an approximation may be measured as follows. +The presence of~$u$ in~\Eq{(1)} lends a certain degree of indeterminateness +to the increase of~$y$: all that we can say (the increase of~$u$ being +known to lie between certain limits) is that $y$~lies in a certain interval +\[ +\eta_{1} \cleq y \cleq \eta_{2}. +\] + +Now (\Ref{ii.}{§\;8}) we can find an increasing function~$F$ so that +$F(\eta_{1}) \ceq F(\eta_{2})$: if $F$~satisfies this condition, any more slowly increasing +function will do so too. \begin{Result}The slower the increase of~$F$ must +be taken, the rougher the approximation.\end{Result} + +{\Loosen The facts may be stated the other way round. Given~$y$, and a +function~$F$, such that $1 \clt F \clt x$, we can determine an interval +$\eta_{1} \cleq y \cleq \eta_{2}$ such that $F(\eta_{1}) \ceq F(\eta_{2})$. The slower the increase of~$F$, +the larger this interval may be taken; if $F \ceq x$ it vanishes, if $F \ceq 1$ +%[** TN: Hardy's notation for a closed interval; inconsistent, not modernizing] +it may be taken as large as we please. If $F = lx$ it might be $(y^{\delta}, y^{\Delta})$; +if $F = l_{2}x$ it might be} +\[ +e^{(ly)^{\delta}}, \qquad +e^{(ly)^{\Delta}}, +\] +and so on. No logarithmico-exponential form of~$F$, however, can give +an interval as large as~$(\log y, e^{y})$; a function~$F$ such that $F(y) \ceq F(e^{y})$ +must transcend any logarithmico-exponential scale. + +\begin{Remark} +Let us consider the approximations~\Eq{(2)} for~$xe^{x}/lx$. + +\Item{(i)} If $y = e^{x^{u}}$ ($u\sim l$), $y$~lies in the interval $e^{x^{1-\delta}}$, $e^{x^{1+\delta}}$. Since +\[ +ll(e^{x^{1-\delta}}) = (1 - \delta)lx \ceq ll(e^{x^{1+\delta}}) +\] +we may take $F = llx$, or even $F = (llx)^{\Delta}$: but the increase of~$F$ cannot be +taken as large as~$(lx)^{\delta}$. + +\Item{(ii)} If $y = e^{(1+u)x}$ ($u \clt 1$), $y$~lies in the interval $e^{(1-\delta)x}$, $e^{(1+\delta)x}$. Then we +may take $F = (lx)^{\Delta}$, but we cannot take $F = e^{(lx)^{\delta}}$. + +\Item{(iii)} {\Loosen If $y = x^{1+u}e^{x}$ we may, as the reader will easily verify, take $F = e^{(lx)^{\mu}}$, +where $\mu$~is any number less than unity.} +%% -----File: 053.png---Folio 45------- + +Another example of an approximation is given by the formula +\[ +\frac{f(x + a)}{f(x)} = e\left\{u\frac{f'(x)}{f(x)}\right\} \quad (u \sim a). +\] + +If, \eg, $a$~is a constant, +\[ +l\left\{\frac{f(x + a)}{f(x)}\right\} + \sim l\left\{e\left[\frac{f'(x)}{f(x)}\right]\right\}, +\] +and the degree of accuracy of the approximation is great enough to be +measured by the function $F = lx$. +\end{Remark} + +\Section{The approximate solution of equations.} + +\Paragraph{4.} It is often important to obtain an asymptotic solution of an +equation $f(x, y) = 0$, \ie\ to find a function whose increase gives an +approximation to that of~$y$. No very general methods of procedure +can be given, but the kind of methods which may be pursued are +worth illustrating by a few examples. + +\Item{(i)} Suppose that the equation is +\[ +x = y\kappa(y), +\Tag{(1)} +\] +where $y^{-\delta} \clt \kappa \clt y^{\delta}$. If the increase of~$\kappa$ is so slow that $\kappa\{y\kappa(y)\} \ceq \kappa(y)$ +it is clear that +\[ +y \ceq x/\kappa(y) \ceq x/\kappa(x): +\] +and if the increase of~$\kappa$ is slow enough we may have $y \sim x/\kappa(x)$. + +The conditions +\[ +e^{-\Delta\sqrt{ly}} \clt \kappa(y) \clt e^{\Delta\sqrt{ly}}, \qquad +e^{-\delta\sqrt{ly}} \clt \kappa(y) \clt e^{\delta\sqrt{ly}} +\] +are, by the result~(v) of~§\;2, enough to ensure the truth of these +hypotheses; and then $y = ux/\kappa(x)$, where $u \ceq 1$ (or $u \sim 1$) is an +approximate solution of our equation. + +\begin{Remark} +Du~Bois-Reymond has proved that the more elaborate approximations +\[ +y = ux/\{\kappa(x/\kappa)\}, \qquad +y = ux\kappa^{-1/\{1+(x\kappa'/\kappa)\}} +\] +have a wider range of validity: and that more elaborate approximations still +may be constructed valid within the range +\[ +e^{-\Delta(ly)^{1-\delta}} \clt \kappa \clt e^{\Delta (ly)^{1-\delta}}. +\] +\end{Remark} + +The more general equation +\[ +x = y^{m}\kappa(y) +\] +can clearly be reduced to the form considered above by writing~$x^{m}$ for~$x$ +and $\kappa^{m}$ for~$\kappa$. +%% -----File: 054.png---Folio 46------- + +In general, if $x = \phi(y)$, the more rapid the increase of~$\phi$ the more +precisely can we determine the increase of~$y$ as a function of~$x$. Thus if +\[ +x = ye^{y} +\] +we have $lx = y + ly$ and +\[ +y = lx - ly = lx(1 + u), +\] +where $u \sim ly/lx \sim llx/lx$. This is a solution of a much more precise +kind than those considered above. + +\Paragraph{5.} The reader will find it instructive to examine the following +results: + +\begin{Remark} +\Item{(i)} Let +\[ +x = ye^{(ly)^{3/8}}. +\] + +This is an example of the work of~§\;4: and +\[ +y \sim xe^{-(lx)^{3/8}}. +\] + +\Item{(ii)} Let +\[ +x = ye^{(ly)^{5/8}}. +\] + +Here +\begin{align*} +y &\sim xe [-(lx)^{5/8} \{1-(lx)^{-3/8}\}^{5/8}]\\ + &\sim xe\{-(lx)^{5/8} + \tfrac{5}{8}(lx)^{1/4}\}. +\end{align*} + +\Item{(iii)} Let +\[ +x = y^{m}(ly)^{m_{1}}(l_{2}y)^{m_{2}} \dots (l_{r}y)^{m_{r}}. +\] + +Here +\[ +y \sim m^{m_{1}/m} x^{1/m} (lx)^{-m_{1}/m} \dots (l_{r}x)^{-m_{r}/m}. +\] + +\Item{(iv)} Let +\[ +x = e^{y^{2}}ly. +\] + +Here +\[ +y = \sqrt{lx - l_{3}x} + u \quad (u \clt 1). +\] + +\Item{(v)} As an example of another type, Du~Bois-Reymond has considered +the equation +\[ +f(x + y) - f(x) = C, +\] +where $C$~is a positive constant. He finds +\begin{gather*} +y \sim C/f'(x) \quad (f(x) \cgt lx),\\ +y = xe\{Cu/xf'(x)\} \quad (u \sim 1,\ lx \cgt f(x) \cgt llx), +\end{gather*} +and so on: the forms of the solution when $f \ceq lx$, $f \ceq llx$,~\dots\ are exceptional. + +\Item{(vi)} As an example of an approximation pushed to greater lengths let us +take the following result: if +\[ +x = y\, ly, +\] +then +\[ +y = \frac{x}{lx} \left\{1 + \frac{llx}{lx} + \frac{(llx)^{2}}{(lx)^{2}} - \frac{llx}{(lx)^{2}}\right\} + u, +\] +where +\[ +u \ceqq \frac{x(llx)^{3}}{(lx)^{4}}. +\] +\end{Remark} + +\Paragraph{6.} Here we may bring our account of the general theory to a +close. It is a theory that has found, and is finding, a large and +increasing variety of applications in various branches of mathematics: +the nature of some of these applications the reader may glean from +Appendix~II\@. +%% -----File: 055.png---Folio 47------- + + +\Appendix{I.}{General Bibliography.} + +\Author{Du~Bois-Reymond}'s memoirs bearing on the subjects of this tract are: + +\Work Sur la grandeur relative des infinis des fonctions (\textit{Annali di +Matematica}, Serie~2, t.~4, p.~338). + +\Work Théorème général concernant la grandeur relative des infinis +des fonctions et de leurs derivées (\textit{Crelle's Journal}, Bd.~74, S.~294). + +\Work Eine neue Theorie der Convergenz und Divergenz von Reihen +mit positiven Gliedern. \textit{Anhang}: Ueber die Tragweite der +logarithmischen Kriterien (\textit{Crelle's Journal}, Bd.~76, S.~61). + +\Work Ueber asymptotische Werthe, infinitäre Approximationen, und +infinitäre Auflösung von Gleichungen (\textit{Math.\ Annalen}, Bd.~8, +S.~363). Nachtrag zur vorstehenden Abhandlung (\textit{ibid.}, S.~574). + +\Work Notiz über infinitäre Gleichheiten (\textit{Math.\ Annalen}, Bd.~10, +S.~576). + +\Work Ueber die Paradoxen des Infinitärcalcüls (\textit{Math.\ Annalen}, +Bd.~11, S.~149). + +\Work Notiz über Convergenz von Integralen mit nicht verschwindendem +Argument (\textit{Math.\ Annalen}, Bd.~13, S.~251). + +\Work Ueber Integration und Differentiation infinitären Relationen +(\textit{Math.\ Annalen}, Bd.~14, S.~498). + +\Work Ueber den Satz: $\lim f'(x) = \lim f(x)/x$ (\textit{Math.\ Annalen}, +Bd.~16, S.~550). + +See also + +\Author{A. Pringsheim}: Ueber die sogenannte Grenze und die Grenzgebiete +zwischen Convergenz und Divergenz (\textit{Münchener Sitzungsberichte}, +Bd.~26, S.~605). + +\Same Ueber die Du~Bois-Reymond'sche Convergenz-Grenze u.s.w. +(\textit{Münchener Sitzungsberichte}, Bd.~27, S.~303). + +\Same Allgemeine Theorie der Convergenz und Divergenz von Reihen +mit positiven Gliedern (\textit{Math.\ Annalen}, Bd.~35, S.~347). + +\Same Zur Theorie der bestimmten Integrale und der unendlichen +Reihen (\textit{Math.\ Annalen}, Bd.~37, S.~591). + +\Author{J. Hadamard}: Sur les caractères de convergence des séries à termes +positifs et sur les fonctions indéfiniment croissantes (\textit{Acta +Mathematica}, t.~18, p.~319 and p.~421). +%% -----File: 056.png---Folio 48------- + +\Author{S. Pincherle}: Alcune osservazioni sugli ordini d'infinito delle funzioni +(\textit{Memorie della Accademia delle Scienze di Bologna}, Ser.~4, t.~5, +p.~739). + +\Author{E. Borel}: \textit{Leçons sur les fonctions entières}, pp.~111--122. + +\Same \textit{Leçons sur les séries à termes positifs}, pp.~1--50. + +\Same \textit{Leçons sur la théorie de la croissance.} + + +\Appendix[Applications.] +{II.}{A Sketch of Some Applications,\protect\footnotemark\ With References.} +\footnotetext{That is to say of certain regions of mathematical theory in which the notation + and the ideas of the \textit{Infinitärcalcül} may be used systematically with a great gain + in clearness and simplicity.} + +\Section[A.]{Convergence and divergence of series and integrals.} + +\Subsection{(i)}{The logarithmic tests.} The series $\sum u_{n}$ ($u_{n} \geq 0$) is convergent if +\begin{flalign*} +&&u_{n} &\cleq n^{-1-\alpha} && \\ +\RTag{\emph{or}} +&&u_{n} &\cleq (n\, ln \dots l_{k-1}n)^{-1}(l_{k}n)^{-1-\alpha}, && +\end{flalign*} +where $\alpha > 0$, and divergent if +\begin{flalign*} +&&u_{n} &\cleq n^{-1} && \\ +\RTag{\emph{or}} +&&u_{n} &\cgeq (n\, ln \dots l_{k}n)^{-1}(l_{k}n)^{-1}. && +\end{flalign*} + +The integral $\ds\int^{\infty} f(x)\,dx$ ($f \geqq 0$) is convergent if +\begin{flalign*} +&&f &\cleq x^{-1-\alpha} && \\ +\RTag{\emph{or}} +&&f &\cleq (x\, lx \dots l_{k-1}x)^{-1}(l_{k}x)^{-1-\alpha}, && +\end{flalign*} +where $\alpha > 0$, and divergent if +\begin{flalign*} +&&f &\cleq x^{-1} && \\ +\RTag{\emph{or}} +&&f &\cleq (x\, lx \dots l_{k}x)^{-1}. && +\end{flalign*} + +The integral $\ds\int_{0} f(x)\,dx$ ($f \geqq 0$) is convergent if +\begin{flalign*} +&&f &\cleq (1/x)^{1-\alpha} && \\ +\qquad{\emph{or}} +&&f &\cleq (1/x)\{l(1/x) \dots l_{k-1}(1/x)\}^{-1} \{l_{k}(1/x)\}^{-1-\alpha}, && +\end{flalign*} +where $\alpha > 0$, and divergent if +\begin{flalign*} +&&f &\cgeq 1/x && \\ +\qquad{\emph{or}} +&&f &\cgeq (1/x)\{l(1/x) \dots l_{k}(1/x)\}^{-1}. && +\end{flalign*} +%% -----File: 057.png---Folio 49------- + +[The first general statement of the `logarithmic criteria,' so far +as series are concerned, appears to have been made by De~Morgan: +see his \textit{Differential and Integral Calculus}, 1839, p.~326. The +essentials of the matter, however, appear in a posthumous memoir +of Abel (\textit{\OE uvres complètes}, t.~2, p.~200; see also t.~1, p.~399). This +memoir appears also to have been first published in 1839. The case +of $k = 1$ had been dealt with by Cauchy (\textit{Exercices de Mathématiques}, +t.~2, 1827, pp.~221 \textit{et~seq.}). Bertrand appears to have arrived at +some or all of De~Morgan's results independently (see \textit{Liouville's Journal}, +t.~7, 1842, p.~37) and the criteria are very commonly attributed to him. +The criteria for integrals do not appear to have been stated generally +before Riemann, \textit{Inaugural-Dissertation} of 1854 (\textit{Werke}, S.~229). + +The following references may also be useful: + +Bonnet, \textit{Liouville's Journal}, t.~8, p.~78. + +Dini, \textit{Sulle serie a termini positivi} (Pisa, 1867); also in the +\textit{Annali dell' Univ.\ Tosc.}, t.~9, p.~41. + +Du~Bois-Reymond, \textit{Crelle's Journal}, Bd.~76, S.~619. + +Pringsheim, \textit{Math.\ Annalen}, Bd.~35, S.~347 and Bd.~37, S.~591; +also in the \textit{Encyklopädie der Math.\ Wiss.}, Bd.~1, Th.~1, S.~77 \textit{et~seq.} + +Hobson, \textit{Theory of functions of a real variable}, p.~406. + +Bromwich, \textit{Infinite series}, pp.~29,~37. + +Hardy, \textit{Course of pure mathematics}, pp.~357 \textit{et~seq.} + +Chrystal, \textit{Algebra}, vol.~2, pp.~109 \textit{et~seq.}] + +\Subsection{(ii)}{General theorems analogous to Du~Bois-Reymond's Theorem +\(\Ref{ii.}{§\;1}\).} + +Given any divergent series $\sum u_{n}$ of positive terms, we can find a +function~$v_{n}$ such that $v_{n} \clt u_{n}$ and $\sum v_{n}$ is divergent; \ie\ given any +divergent series we can find one more slowly divergent. + +Given any convergent series $\sum u_{n}$ of positive terms we can find~$v_{n}$ +so that $v_{n} \cgt u_{n}$ and $\sum v_{n}$ is convergent; \ie\ given any convergent +series we can find one more slowly convergent. + +Given any function~$\phi(n)$ tending to infinity, however slowly, we +can find a convergent series~$\sum u_{n}$ and a divergent series~$\sum v_{n}$ such +that $v_{n}/u_{n} = \phi(n)$. + +Given an infinite sequence of series, each converging (diverging) +more slowly than its predecessor, we can find a series which converges +(diverges) more slowly than any of them. + +[See Abel and Dini, \lc~\textit{supra}; Hadamard, \textit{Acta Mathematica}, t.~18, +p.~319 and t.~27, p.~177; Bromwich, \textit{Infinite series}, p.~40; Littlewood, +\textit{Messenger of Mathematics}, vol.~39, p.~191.] +%% -----File: 058.png---Folio 50------- + +There is no function~$\phi(n)$ such that $u_{n}\phi(n) \cgeq 1$ is a necessary +condition for the divergence of $\sum u_{n}$, and no function~$\phi(n)$ such that +$\phi(n) \cgt 1$ and $u_{n}\phi(n) \cleq 1$ is a necessary condition for the convergence +of~$\sum u_{n}$. + +If $u_{n}$~is a \emph{steadily decreasing} function of~$n$, then $nu_{n} \clt 1$ \emph{is} a +necessary condition for convergence; but there is no function~$\phi(n)$ +such that $\phi(n) \cgt 1$ and $n\phi(n) u_{n} \clt 1$ is a necessary condition. + +[Pringsheim, \textit{Math.\ Annalen}, Bd.~35, S.~343 \textit{et~seq.; ibid.}, Bd.~37, +S.~591 \textit{et~seq.}] + +If however $nu_{n}$~decreases steadily, then $n\log nu_{n} \to 0$ is a necessary +condition; and if $n\psi(n)u_{n}$, where $n\psi(n) \cgt 1$ and $\ds\int \frac{dn}{n\psi(n)} \cgt 1$, decreases +steadily, then +\[ +\left(n\psi(n) \int \frac{dn}{n\psi(n)}\right) u_{n} \to 0 +\] +is a necessary condition. + +\Subsection{(iii)}{Special series and integrals possessing peculiarities in respect +to the mode of their convergence or divergence.} + +For examples of series and integrals which converge or diverge +so slowly as not to answer to any of the logarithmic criteria see +Du~Bois-Reymond, Pringsheim, Borel (\lc~\textit{supra}), and Blumenthal, +\textit{Principes de la théorie des fonctions entières d'ordre infini}, ch.~1. + +In these cases the logarithmic tests are insufficient (\textit{en~défaut}, +\Ref{iv.}{§§\;2,~5}). For examples of series and integrals to which the +logarithmic tests are \emph{inapplicable} (\Ref{v.}{§§\;3,~4}) see the writings just +mentioned and also + +Thomae: \textit{Zeitschrift für Mathematik}, Bd.~23, S.~68. + +Gilbert: \textit{Bulletin des Sciences Mathématiques}, t.~12, p.~66. + +Goursat: \textit{Cours d'Analyse}, t.~1, p.~205. + +Hardy: \textit{Messenger of Mathematics}, vol.~31, p.~1; \textit{ibid.},~vol.~31, +p.~177; \textit{ibid.},~vol.~39, p.~28. + +\Section[B.]{Asymptotic formulae for finite series and integrals.} + +A closely connected problem is that of the determination of +asymptotic formulae for +\[ +A_{n} = a_{1} + a_{2} + \dots + a_{n} +\] +or for +\[ +\Phi(x) = \int_{a}^{x} \phi(t)\,dt, +\] +{\Loosen when the behaviour of $a_{n}$ or~$\phi(x)$ for large values of $n$ or~$x$ is known. +A good deal can be accomplished in this direction by means of +%% -----File: 059.png---Folio 51------- +(i)~the theorem of Cauchy and Stolz, that, if $a_{n}$~and~$b_{n}$ are positive and +$a_{n} \sim Cb_{n}$, then $A_{n} \sim CB_{n}$, (ii)~the theorems of~\Ref{vi.}{}\ and (iii)~the theorem +of Maclaurin and Cauchy, that} +\[ +\phi(1) + \phi(2) + \dots + \phi(n) - \int_{1}^{n} \phi(x)\,dx, +\] +where $\phi(x)$~is a positive and decreasing function of~$x$, tends to a limit +as $n \to \infty$. + +[For~(i) see Cauchy, \textit{Analyse algébrique}, p.~52; Stolz, \textit{Math.\ +Annalen}, Bd.~14, S.~232, or \textit{Allgemeine Arithmetik}, Bd.~1, S.~173; +Jensen, \textit{Tidskrift for Mathematik}~(5), Bd.~2, S.~81; Bromwich, +\textit{Infinite series}, p.~378, and \textit{Proc.\ Lond.\ Math.\ Soc.}, ser.~2, vol.~7, +p.~101. Proofs of~(iii) will be found in almost any modern treatise +on analysis: \eg, Bromwich, \textit{Infinite series}, p.~29; Hardy, \textit{Course +of pure mathematics}, p.~305. An important extension to \emph{slowly +oscillating} series has been given recently by Bromwich (\textit{Proc.\ Lond.\ +Math.\ Soc.}, ser.~2, vol.~7, p.~327).] + +Among the most important results which follow from these +theorems are +\begin{gather*} +1^{s} + 2^{s} + \dots + n^{s} \sim \frac{n^{s+1}}{s + 1} \quad (s > -1), \\ +1^{s} + 2^{s} + \dots + n^{s} - \frac{n^{s+1}}{s + 1} \sim \zeta(-s) \quad (-1 < s < 0), \\ +1 + \frac{1}{2} + \dots + \frac{1}{n} - \log n \sim \gamma, +\end{gather*} +\begin{multline*} +1 + \frac{\alpha·\beta}{1·\gamma} + + \frac{\alpha(\alpha + 1) \beta(\beta + 1)}{1·2·\gamma(\gamma + 1)} + \dots\ + \text{to $n$~terms,} \\ +\begin{aligned} + &\sim \frac{\Gamma(\gamma)}{\Gamma(\alpha)\, \Gamma(\beta)}\, + \frac{n^{\alpha+\beta-\gamma}}{\alpha + \beta - \gamma}\quad + (\alpha + \beta > \gamma), \\ +\LTag{\emph{or}} + &\sim \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\, \Gamma(\beta)}\, + \log n\quad + (\alpha + \beta = \gamma). +\end{aligned} +\end{multline*} + +In connection with the last result see Bromwich, \textit{Proc.\ Lond.\ Math.\ +Soc.}, ser.~2, vol.~7, p.~101; in the earlier formulae $\gamma$~is Euler's constant +and $\zeta$~denotes the `Riemann $\zeta$-function.' + +The most important of all formulae of this kind is beyond question +\[ +\log 1 + \log 2 + \dots + \log n - (n + \tfrac{1}{2})\log n + n \sim \tfrac{1}{2} \log(2\pi), +\] +which, in the form +\[ +n! \sim n^{n+\frac{1}{2}} e^{-n} \sqrt{2\pi}, +\] +constitutes \emph{Stirling's Theorem}. The literature connected with Stirling's +Theorem and its extensions to the Gamma-function of a non-integral +%% -----File: 060.png---Folio 52------- +or complex variable is far too extensive to be summarized here. See +\textit{Encykl.\ der Math.\ Wiss.}, Bd.~II.~(2), S.~165 \textit{et~seq.}; Bromwich, \textit{Infinite +series}, pp.~461 \textit{et~seq.} + +Another formula of the same kind is +\[ +1^{1}2^{2}3^{3} \dots n^{n} \sim An^{\frac{1}{2}n^{2} + \frac{1}{2}n + \frac{1}{12}}\, e^{-\frac{1}{4}n^{2}}, +\] +where $A$~is a constant defined by the equation +\[ +\log A = \tfrac{1}{12}\log 2\pi + \tfrac{1}{12} \gamma + + \frac{1}{2\pi^{2}} \sum_{1}^{\infty} \frac{\log \nu}{\nu^{2}}. +\] + +The properties of this constant have been investigated by Kinkelin +and Glaisher (Kinkelin, \textit{Crelle's Journal}, Bd.~57, S.~122: Glaisher, +\textit{Messenger of Mathematics}, vol.~6, p.~71; vol.~7, p.~43; vol.~23, p.~145; +vol.~24, p.~1; \textit{Quarterly Journal of Mathematics}, vol.~26, p.~1: see also +Barnes, \textit{ibid.},~vol.~31, pp.~264 \textit{et~seq.}). + +All these results are intimately bound up with the theory of +the general `Euler-Maclaurin Sum Formula' +\[ +\sum_{1}^{n} f(n) + = \int^{n} f(x)\,dx + C + \tfrac{1}{2}f(n) + \frac{B_{1}}{2!} f'(n) - \frac{B_{2}}{4!} f'''(n) + \dots +\] +which also possesses an extensive literature (see Schlömilch, \textit{Theorie +der Differenzen und Summen}; Boole, \textit{Finite differences}; Markoff, +\textit{Differenzenrechnung}; Seliwanoff, \textit{Differenzenrechnung}; \textit{Encykl.\ der +Math.\ Wiss.}, Bd.~I. S.~929 \textit{et~seq.}; Bromwich, \textit{Infinite series}, +p.~238 and p.~324; Barnes, \textit{Proc.\ Lond.\ Math.\ Soc.}, ser.~2, vol.~3, +pp.~253 \textit{et~seq.}; where many further references are given). + +A simple example of the use of the general formula is afforded +by the relation +\[ +\sum_{1}^{n} \nu^{s} - \frac{n^{s+1}}{s + 1} - \tfrac{1}{2} n^{s} + - \sum_{1} (-1)^{i-1} \left(\frac{s}{2i - 1}\right) \frac{B_{i}}{2i} n^{s-2i+1} \sim \zeta(-s). +\] + +Here $s$~is positive and not integral, and the summation with +respect to~$i$ is continued until we come to a negative power of~$n$. + +\Section[C.]{Formulae involving prime numbers only.} + +Asymptotic formulae involving functions defined arithmetically, +and particularly functions defined by sums of functions of prime +numbers only, play a most important part in the analytical theory +of numbers. Of these the most important is the formula +\[ +\Pi(n) \sim \frac{n}{ln}, +\] +where $\Pi(n)$~denotes the number of prime numbers less than~$n$. +%% -----File: 061.png---Folio 53------- + +Similarly it is known that +\[ +\sum lp \sim n, \qquad +\sum \frac{lp}{p} \sim ln, \qquad +\sum \frac{1}{p} \sim lln +\] +(the summation in each case applying to all primes less than~$n$) while +$\sum\limits^{\infty} \dfrac{1}{p\, lp}$ is convergent. + +Many more accurate results have been established by recent +writers, particularly Mertens, Hadamard, Von~Mangoldt, De~la~Vallée-Poussin, +and Landau; and the theory has to a considerable extent +been freed from Riemann's still unproved assumption that all the +roots of his Zeta-function have their real part equal to~$\frac{1}{2}$. Thus it +has been shown that +\[ +\Pi(n) = \int_{2}^{n} \frac{dx}{\log x} + O\left\{\frac{n}{(ln)^{\Delta}}\right\}, +\] +or, still more accurately, +\[ +\Pi(n) = \int_{2}^{n} \frac{dx}{\log x} + O\{ne^{-\alpha\sqrt{ln}}\}, +\] +where $\alpha$~is a positive constant; but it still remains to be settled +whether (as there is some reason to suppose) the last term can be +replaced by~$O(\sqrt{n})$ or even by +\[ +O\left(\frac{\sqrt{n}}{ln}\right). +\] + +[It would carry us too far to give detailed references to the +literature of this exceedingly difficult and fascinating subject. The +reader should consult Landau's exhaustive \textit{Handbuch der Lehre von +der Verteilung der Primzahlen} (Teubner, 1909).] + +\Section[D.]{The theory of integral functions.} + +\Subsection{1.}{}The series $\sum c_{n}x^{n}$ will converge for all values of~$x$ (real or +complex), and so define an \emph{integral function}~$f(x)$, if and only if +$\sqrt[n]{|c_{n}|} \to 0$, \ie\ if $|c_{n}| \clt e^{-\Delta n}$. + +\Subsection{2.}{The three indices of a function of finite order.} The three +most important characters of an integral function~$f(x)$ are: + +\Item{(i)} $\gamma_{n} = |c_{n}|$, the modulus of the $n$th~coefficient; + +\Item{(ii)} $\alpha_{n} = |a_{n}|$, the modulus of the $n$th (in order of absolute +magnitude) zero of~$f(x)$; + +\Item{(iii)} $M(r)$, the maximum of~$|f(x)|$ on the circle $|x| = r$. $M(r)$~is +known to be an increasing function of~$r$, and in all cases $M(r) \cgt r^{\Delta}$. +%% -----File: 062.png---Folio 54------- + +A function such that $M(r) \clt e^{r^{\Delta}}$ is called a \emph{function of finite +order}. We shall confine our remarks to such functions. + +The principal problem of the theory of integral functions is the +determination of the relations between the increases of $\alpha_{n}$,~$1/\gamma_{n}$, and~$M(r)$. +Those which subsist between the two latter functions are the +simplest: when $\alpha_{n}$~is taken into account the theory is complicated by +the `Picard case of exception'---the case of functions which (like~$e^{x}$) +have no zeroes, or whose zeroes are scattered abnormally widely over +the plane. + +The nature of the results of the general theory may be gathered +from a statement of a few of the simplest of them. + +If +\[ +n^{-\mu-\delta} \clt \sqrt[n]{\gamma_{n}} \clt n^{-\mu+\delta}, +\] +\ie\ if +\[ +l(1/\gamma_{n}) \sim \mu n\, ln, +\] +we call $\mu$ the \emph{$\mu$-index}. The index may be defined in \emph{all} cases without +any assumption as to the existence of a limit for $\{l(1/\gamma_{n})/(n\, ln)\}$; we +confine ourselves to the simplest case. + +If +\[ +n^{(1/\lambda)-\delta} \clt \alpha_{n} \clt n^{(1/\lambda)+\delta}, +\] +we call $\lambda$ the \emph{$\lambda$-index}; and if +\[ +e^{r^{\nu-\delta}} \clt M(r) \clt e^{r^{\nu+\delta}}, +\] +we call $\nu$ the \emph{$\nu$-index}: thus +\[ +l\alpha_{n} \sim (ln)/\lambda, \qquad +llM(r) \sim \nu\, lr. +\] + +Then $\mu = 1/\nu$: and \emph{in general} $\lambda = \nu$. + +Thus for the function +\[ +\frac{\sin(\sqrt{x})}{\sqrt{x}} = 1 - \frac{x}{3!} + \frac{x^{2}}{5!} - \dots +\] +we have $\lambda = \nu = \frac{1}{2}$ and $\mu = 2$, as the reader will easily verify (using +Stirling's Theorem to determine~$\mu$). + +\Subsection{3.}{Special results.} More precise results than these have been +obtained in many cases. Thus if +\[ +\{n(ln)^{-\alpha_{1}} \dots (l_{\nu}n)^{-\alpha_{\nu}+\delta}\}^{-1/\rho} + \clt \sqrt[n]{\gamma_{n}} + \clt \{n(ln)^{-\alpha_{1}} \dots (l_{\nu}n)^{-\alpha_{\nu}-\delta}\}^{-1/\rho}, +\] +then +\[ +e\{r^{\rho}(lr)^{\alpha_{1}} \dots (l_{\nu}r)^{\alpha_{\nu}-\delta}\} + \clt M(r) + \clt e\{r^{\rho}(lr)^{\alpha_{1}} \dots (l_{\nu}r)^{\alpha_{\nu}+\delta}\}, +\] +and conversely. +%% -----File: 063.png---Folio 55------- + +As examples of still more accurate, but more special results, we +may quote the following: +\begin{align*} +&\sum \frac{x^{n}}{n^{\alpha n}} + \sim \sqrt{\frac{2\pi}{e\alpha}}\, x^{1/2\alpha} e^{(\alpha/e)x^{1/\alpha}},\\ +&\sum \frac{x^{n}}{(n!)^{\alpha}} + \sim \frac{1}{\sqrt{\alpha}}\, + (2\pi)^{(1-\alpha)/2} x^{(1-\alpha)/2\alpha} e^{\alpha x^{1/\alpha}},\\ +&\sum \frac{x^{n}}{\Gamma(\alpha n + 1)} \sim (1/\alpha) e^{x^{1/\alpha}},\\ +&\sum e^{-n^{p}}x^{n} +% [** TN: Braces (not parentheses) in sqrt in original] + \sim \sqrt{\frac{2\pi}{p(p - 1)}} \left(\frac{\log x}{p}\right)^{\frac{2-p}{2p-2}} + e^{(p-1)\left(\frac{\log x}{p}\right)^{p/(p-1)}}, +\end{align*} +where $\alpha > 0$ and in the last formula $1 < p < 2$, and throughout $x \to \infty$ +by positive values. + +These results may of course be used to give an upper limit for the +modulus of the particular function considered when $x$~is not necessarily +real, and so for~$M(r)$. Thus in the first case +\[ +M(r) \cleq r^{1/2\alpha} e^{(\alpha/e) x^{1/\alpha}}. +\] + +[The reader who wishes to become familiar with the theory of +integral functions should begin by reading Borel's \textit{Leçons sur les +fonctions entières}. Some additions will be found in the notes at the +end of the same writer's \textit{Leçons sur les fonctions méromorphes}. He +should then read two memoirs by E.~Lindelöf; a short one in the +\textit{Bulletin des Sciences Mathématiques}, t.~27, p.~1, and a long one in +the \textit{Acta Societatis Fennicae}, t.~31, p.~1. Some of the results of this +last paper were proved independently by Boutroux (\textit{Acta Mathematica}, +t.~28, pp.~97 \textit{et~seq.}); but M.~Boutroux's important memoir is largely +occupied by a discussion of some of the most difficult points in the +theory. + +Much of the theory has been developed in a very simple and +elementary way by Pringsheim (\textit{Math.\ Annalen}, Bd.~58, S.~257); and +the reader should certainly consult a short note by Le~Roy (\textit{Bulletin +des Sciences Mathématiques}, t.~24, p.~245). But, after reading the +works of Borel and Lindelöf mentioned above, he will be wise to turn +to Vivanti's \textit{Teoria delle funzioni analitiche} (German translation by +Gutzmer), which contains by far the fullest treatment of the subject +yet published, and an exhaustive list of original memoirs.] +%% -----File: 064.png---Folio 56------- + +\Section[E.]{Power series with a finite radius of convergence.} + +Suppose that $a_{1} + a_{2} + \dots$ is a divergent series: for simplicity +suppose that $a_{n}$~is always positive and steadily increases or decreases +as $n$~increases. Further suppose $e^{-\delta n} \clt a_{n} \clt e^{\delta n}$, so that $\sum a_{n}x^{n}$ is +convergent if $0 \leqq x < 1$. Then a large number of interesting results +have been established connecting the increase of~$a_{n}$, as $n \to \infty$, and +that of $f(x) = \sum a_{n}x^{n}$ as $x \to 1$. The fundamental result is: \emph{if $a_{n} \sim Cb_{n}$, +or, more generally, if $(a_{1} + a_{2} + \dots + a_{n}) \sim C(b_{1} + b_{2} + \dots + b_{n})$, and +$f(x) = \sum a_{n}x^{n}$, $g(x) = \sum b_{n}x^{n}$, then} +\[ +f(x) \sim Cg(x). +\] + +From this theorem it may be deduced that +\begin{align*} +\sum \frac{x^{n}}{n^{p}} + &\sim \frac{\Gamma(1 - p)}{(1 - x)^{1-p}}\quad (p < 1), \\ +F(\alpha, \beta, \gamma, x) + &\sim \frac{\Gamma(\gamma)\, \Gamma(\alpha + \beta - \gamma)}{\Gamma(\alpha)\, \Gamma(\beta)}\, + \frac{1}{(1 - x)^{\alpha+\beta-\gamma}}\quad (\alpha + \beta > \gamma) \\ +F(\alpha, \beta, \alpha + \beta, x) + &\sim \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\, \Gamma(\beta)}\, + l\left(\frac{1}{1 - x}\right). +\end{align*} + +Of further results the following is typical: if +\[ +a_{n} \sim n^{p}/n\, ln \dots l_{m-1}n (l_{m}n)^{q} \dots (l_{m+k}n)^{q_{k}}, +\] +then +\begin{multline*} +F(x) \sim \Gamma(p) \bigg/ \bigg\{(1 - x)^{p+1} \\ + × \frac{1}{1 - x}\, l\frac{1}{1 - x} \dots l_{m-1} \frac{1}{1 - x} + \left(l_{m} \frac{1}{1 - x}\right)^{q}\!\! \dots + \left(l_{m+k} \frac{1}{1 - x}\right)^{q_{k}}\bigg\} +\end{multline*} +if $p > 0$, $q \neq 1$: but +\[ +F(x) \sim 1 \bigg/ + \biggl\{(1 - q)\left(l_{m} \frac{1}{1 - x}\right)^{q-1}\!\! + \biggl(l_{m+1} \frac{1}{1 - x}\biggr)^{q_{1}}\!\! \dots + \biggl(l_{m+k} \frac{1}{1 - x}\biggr)^{q_{k}}\biggl\} +\] +if $p = 0$, $q < 1$ (if $p < 0$ or $p = 0$, $q > 1$, then $\sum a_{n}$ is convergent). + +Thus, \eg +\[ +\sum \frac{n^{p} x^{n}}{(lx)^{q}} + \sim \Gamma(p + 1) \bigg/ + \left\{(1 - x)^{p+1} \left(l \frac{1}{1 - x}\right)^{q}\right\}. +\] +%% -----File: 065.png---Folio 57------- + +As specimens of further results of this character we may quote +\begin{gather*} +x + x^{4} + x^{9} + \dots \sim \tfrac{1}{2} \sqrt{\frac{\pi}{1 - x}}, \\ +x + x^{\alpha} + x^{\alpha^{2}} + \dots + \sim \frac{1}{la}\, l\left(\frac{1}{1 - x}\right)\quad (a > 1), \\ +\sum a^{n} x^{n^{2}} + \sim e\left\{\tfrac{1}{4}\, \frac{(la)^{2}}{l(1/x)}\right\} \quad (a > 1), \\ +\sum e^{n/ln} x^{n} = e_{2}\{u/(1 - x)\} \quad (u \sim 1). +\end{gather*} +Many similar results have been established about series other than +power series: thus +\begin{align*} +\sum \frac{x^{n}}{n(1 + x^{n})} + &\sim \tfrac{1}{2}\, l\left(\frac{1}{1 - x}\right), \\ +\sum \frac{x^{n}}{1 - x^{n}} + &\sim \frac{1}{1 - x}\, l\left(\frac{1}{1 - x}\right). +\end{align*} +As an example of a more precise result we may quote the formula +\[ +\sum \frac{x^{n}}{1 + x^{2n}} + = \tfrac{1}{4} \left\{\frac{\pi}{l(1/x)} - 1\right\} + O\{(1 - x)^{\Delta}\}. +\] +[See + +Bromwich, \textit{Infinite series}, pp.~131 \textit{et~seq.}, 171~\textit{et~seq.}; + +Le~Roy, \textit{Bulletin des Sciences Mathématiques}, t.~24, pp.~245 \textit{et~seq.}; + +Lasker, \textit{Phil.\ Trans.\ Roy.\ Soc.},~(A), vol.~196, p.~433; + +Pringsheim, \textit{Acta Mathematica}, t.~28, p.~1; + +Barnes, \textit{Proc.\ Lond.\ Math.\ Soc.}, vol.~4, p.~284; \textit{Quarterly Journal}, +vol.~37, p.~289; + +Hardy, \textit{Proc.\ Lond.\ Math.\ Soc.}, vol.~3, p.~381; \textit{ibid.},~vol.~5, p.~197; +\textit{ibid.},~vol.~5, p.~342; \\ +where further references will be found. These writers also consider +the extensions of such results to the field of the complex variable.] +%% -----File: 066.png---Folio 58------- + +\Appendix{III.}{Some Numerical Illustrations.} + +Mr J.~Jackson, scholar of Trinity College, has been kind enough to +calculate for me the following numerical results, which will, I think, +be found instructive as comments on some of the matters dealt with in +the body of this tract and in Appendix~II\@. It will of course be understood +that, except in one or two instances, they are approximations +and sometimes quite rough approximations. + +\Section[1.]{Table of the functions $\log x$, $\log\log x$, $\log\log\log x$, etc.} +{ +\[ +\begin{array}{|l|r|r|r|r|r|} +\hline +\CCEntry{x} & \CEntry{\log x} & \CEntry{\log_{2} x} & \CEntry{\log_{3} x} & \CEntry{\log_{4} x} & \CEntry{\log_{5} x}\\ +\hline +\Strut +10 & 2.30 & 0.834 & -0.182 & \Dash & \Dash\\ +10^{3} & 6.91 & 1.933 & 0.659 & -0.417 & \Dash\\ +10^{6} & 13.82 & 2.626 & 0.966 & -0.035 & \Dash\\ +10^{10} & 23.03 & 3.137 & 1.143 & 0.134 & -2.011\\ +10^{15} & 34.54 & 3.542 & 1.265 & 0.235 & -1.449\\ +10^{20} & 46.05 & 3.830 & 1.343 & 0.295 & -1.221\\ +10^{30} & 69.08 & 4.235 & 1.443 & 0.367 & -1.003\\ +10^{60} & 138.15 & 4.928 & 1.595 & 0.467 & -0.762\\ +10^{100} & 230.26 & 5.439 & 1.693 & 0.527 & -0.641\\ +10^{1000} & 2302.58 & 7.742 & 2.047 & 0.716 & -0.334\\ +10^{10^{6}} & 2303 × 10^{3} & 14.650 & 2.685 & 0.987 & -0.013\\ +10^{10^{10}} & 2303 × 10^{7} & 23.860 & 3.172 & 1.154 & 0.144\\ +\hline +\end{array} +\] +} +%% -----File: 067.png---Folio 59------- + +\Section[2.]{Table of the functions $e^{x}$, $e^{e^{x}}$, $e^{e^{e^{x}}}$, etc.} +\[ +\begin{array}{|c|c|c|c|c|c|} +\hline +\CCEntry{x} & \CEntry{ex} & \CEntry{e_{2}x} & \CEntry{e_{3}x} & \CEntry{e_{4}x} \\ +\hline +\Strut +1 & 2.718 & 15.154 & 3,814,260 & 10^{1,656,510}\\ +2 & 7.389 & 1618.2 & 5.85 × 10^{702} & \Dash\\ +3 & 20.085 & 5.28 × 10^{8} & 10^{2.295 × 10^{8}} & \Dash\\ +5 & 148.413 & 2.85 × 10^{64} & 10^{1.24 × 10^{64}} & \Dash\\ +10 & 22026 & 9.44 × 10^{9565} & \Dash & \Dash\\ +\hline +\end{array} +\] + +The function $\log x$ is defined only for $x > 0$, $\log_{2}x$~for $x > 1$, +$\log_{3}x$~for $x > e$, $\log_{4} x$~for $x > e^{e} = e_{2}$, and so on. The values of the +first few numbers $e$,~$e_{2}$, $e_{3}$,~\dots\ are given above, viz.\ $e = 2.718$, $e_{2} = 15.154$, +$e_{3} = 3,814,260$, $e_{4} = 10^{1,656,510}$. + +\Section[3.]{Table of the functions $n!$, $n^{n}$, $n^{n^{n}}$.} + +\[ +\begin{array}{|c|c|c|c|} +\hline +\CCEntry{n} & \CEntry{n!} & \CEntry{n^{n}} & \CEntry{n^{n^{n}}} \\ +\hline +\Strut +1 & 1 & 1 & 1\\ +2 & 2 & 4 & 16\\ +3 & 6 & 27 & 7.634 × 10^{12}\\ +4 & 24 & 256 & 1.491 × 10^{154}\\ +5 & 120 & 3,125 & 9.55 × 10^{2,184}\\ +6 & 720 & 46,656 & 2.7 × 10^{36,305}\\ +7 & 5,040 & 823,543 & 1.4 × 10^{695,974}\\ +8 & 40,320 & \DPtypo{16,827,216}{16,777,216} & 10^{15,151,345}\\ +9 & 362,880 & 3.8742 × 10^{8} & 10^{369,693,100}\\ +10 & 3,628,800 & 10^{10} & 10^{10,000,000,000}\\ +100 & 9.346 × 10^{157} & 10^{200} & \Dash\\ +10^{10} & 10^{9.57 × 10^{10}} & 10^{10^{11}} & \Dash\\ +\hline +\end{array} +\] +%% -----File: 068.png---Folio 60------- + +\Section[4.]{Table to illustrate the convergence of the series\DPtypo{}{.}} + +{\small +\begin{gather*} +\begin{aligned} +&(1)\ \sum_{3}^{\infty} \frac{1}{n\log n (\log\log n)^{2}}. +&&(2)\ \sum_{2}^{\infty} \frac{1}{n(\log n)^{2}}. +&&(3)\ \sum_{1}^{\infty} \frac{1}{n^{s}}\ (s = 1.1).\\ +&(4)\ \sum_{1}^{\infty} \frac{1}{n^{s}}\ (s = 1.5). +&&(5)\ \sum_{1}^{\infty} \frac{1}{n^{s}}\ (s = 2). +&&(6)\ \sum_{1}^{\infty} \frac{1}{n^{s}}\ (s = 10).\\ +&(7)\ \sum_{1}^{\infty} \frac{1}{n^{s}}\ (s = 100). +&&(8)\ \sum_{0}^{\infty} x^{n}\ (x = .9). +&&(9)\ \sum_{0}^{\infty} x^{n}\ (x = .5). +\end{aligned} \\ +\begin{aligned} +&(10)\ \sum_{0}^{\infty} x^{n}\ (x = .1). +&&(11)\ 1 + \frac{1}{2!} + \frac{1}{3!} + \dots. +&&(12)\ 1 + \frac{1}{2^{2}} + \frac{1}{3^{3}} + \dots.\\ +&(13)\ \sum_{0}^{\infty} x^{n^{2}}\ (x = .9). +&&(14)\ \sum_{0}^{\infty} x^{n^{2}}\ (x = .5) +&&(15)\ \sum_{0}^{\infty} x^{n^{2}}\ (x = .1). +\end{aligned} \\ +(16)\ \frac{1}{1^{1^{1}}} + \frac{1}{2^{2^{2}}} + \frac{1}{3^{3^{3}}} +\dots. +\end{gather*} +\footnotesize\settowidth{\TmpLen}{calculate the sum correctly to}% +\[ +%[** TN: Force centering of slightly over-wise table] +\makebox[0pt][c]{$ +\begin{array}{|c|c|c|c|c|c|} +\hline +&&\multicolumn{4}{|c|}{% + \parbox{\TmpLen}{% + \centering\footnotesize\medskip Number of terms required to\\ + calculate the sum correctly to}} \\ +\text{Series} & \text{Sum} & 2 & 10 & 100 & 1000 \\ +&&&\multicolumn{2}{|c|}{\centering\text{\footnotesize decimal places\footnotemark}} & \\ +\hline +\Strut +1 & 38.43 & 10^{3.14 × 10^{86}} & \Dash & \Dash & \Dash\\ +2 & 2.11 & 7.23 × 10^{86} & 10^{8.6 × 10^{9}} & \Dash & \Dash\\ +3 & 10.58 & 10^{33} & 10^{113} & 10^{1013} & 10^{10013}\\ +4 & 2.612 & 160,000 & 16 × 10^{20} & 6 ×10^{200} & 16 ×10^{2000}\\ +5 & \frac{1}{6}\pi^{2} = 1.64493 & 200 & 2 × 10^{10} & 2 × 10^{100} & 2 × 10^{1000}\\ +6 & 1.0009846 & 1 & 11 & 1.093 × 10^{11} & 1.093 × 10^{111}\\ +7 & 1 + (1.27 × 10^{-30}) & 1 & 1 & 10 & 1.213 × 10^{10}\\ +8 & 10 & 73 & 247 & 2214 & 21883\\ +9 & 2 & 9 & 36 & 336 & 3325\\ +10 & 10/9 & 3 & 11 & 101 & 1001\\ +11 & e - 1 = 1.718282 & 5 & 13 & 70 & 440\\ +12 & 1.291286 & 3 & 10 & 57 & 386\\ +13 & 3.234989 & 8 & 15 & 46 & 148\\ +14 & 1.564468 & 3 & 6 & 19 & 58\\ +15 & 1.100100 & 2 & 4 & 11 & 32\\ +16 & 1.062500 & 2 & 2 & 3 & 4\\ +\hline +\end{array}$} +\]} +\footnotetext{The phrase `calculate the sum correctly to $m$~decimal places' is used as + equivalent to `calculate with an error less than $\frac{1}{2} × 10^{-m}$.' In the case of a very + slowly convergent series the interpretation affects the numbers to a considerable + extent. The numbers would be considerably more difficult to calculate were the + phrase interpreted in its literal sense.}% +%% -----File: 069.png---Folio 61------- + +Such a series as~(7) is of course exceedingly rapidly convergent \emph{at +first}, \ie\ a very few terms suffice to give the sum correctly to a considerable +number of places; but if the sums are wanted to a very large +number of places, even the series~(8) proves to be far more practicable. + +Mr William Shanks (\textit{Proc.\ Roy.\ Soc.}, vol.~21, p.~318) calculated +the value of~$\pi$ to $707$~places of decimals from Machin's formula +\[ +\pi = 16\left(\frac{1}{5} - \frac{1}{3·5^{3}} + \frac{1}{5·5^{5}} - \dots\right) + - 4 \left(\frac{1}{239} - \frac{1}{3·239^{3}} + \dots\right). +\] +He does not state the number of terms he found it necessary to use, +but, in a previous calculation to $530$~places, used $747$~terms of the +first and $219$~terms of the second series. He also (\textit{ibid.}, vol.~6, p.~397) +calculated~$e$ to $205$~places from the series~(11). + + +\Section[5.]{Table to illustrate the divergence of the series} + +{\small +\begin{align*} +&(1)\ \frac{1}{\log \log 3} + \frac{1}{\log \log 4} + \dots. +&&(2)\ \frac{1}{\log 2} + \frac{1}{\log 3} + \dots.\\ +&(3)\ 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots. +&&(4)\ 1 + \frac{1}{2} + \frac{1}{3} + \dots.\\ +&(5)\ \frac{1}{2\log 2} + \frac{1}{3\log 3} + \dots. +&&(6)\ \frac{1}{3\log 3\log\log 3} + \frac{1}{4\log 4 \log\log 4} \DPtypo{}{{}+ \dots}. +\end{align*} +\[ +\begin{array}{|c|c|c|c|c|c|c|} +\hline +&\multicolumn{6}{|c|}{% + \text{\footnotesize Number of terms required to make the sum greater than}} \\ +\text{Series} & 3 & 5 & 10 & 100 & 1000 & 10^{6}\\ +\hline +\Strut +1 & 1 & 1 & 1 & 116 & 1800 & 2.6 × 10^{6}\\ +2 & 3 & 7 & 20 & 440 & 7600 & 1.5 × 10^{7}\\ +3 & 5 & 10 & 33 & 2500 & 2.5 × 10^{5} & 2.5 × 10^{11}\\ +4 & 11 & 82 & 12390 & 10^{43} & 10^{.43 ×10^{3}} & 10^{.43 × 10^{6}}\\ +5 & 8690 & 1.3 × 10^{29} & 10^{4300} & 10^{5 × 10^{42}} & \Dash & \Dash\\ +6 & 1 & 60 \text{ \emph{to} } 70 & 10^{10^{100}} & \Dash & \Dash & \Dash\\ +\hline +\end{array} +\]} + +\Section[6.]{Roots of certain equations.} + +\Item{(i)} The equation $e^{x} = x^{1,000,000}$ has a root just larger than unity (the +excess of the root over unity being practically~$10^{-6}$) and a large root +in the neighbourhood of~$16,610,800$. The equation $e^{x} = 1,000,000 x^{1,000,000}$ +has roots nearly equal to those of the above. The one near unity is +practically $12.82 x 10^{-6}$ less than unity, while the large root exceeds +the root of the above equation by about~$13.82$. +%% -----File: 070.png---Folio 62------- + +\Item{(ii)} The equation $e^{x^{2}} = x^{10^{10}}$ has a root somewhere near~$357,500$. + +\Item{(iii)} {\Loosen The equation $e^{e^{x}} = 10^{10} x^{10} e^{10^{10} x^{10}}$ has a root near~$64.7$. The +root differs by less than~$10^{-26}$ from the corresponding root of $e^{x} = 10^{10} x^{10}$. +The corresponding root of $e^{x} = x^{10}$ is about~$35.8$.} + +\Item{(iv)} The positive roots of $x^{x^{x}} = 1,000,000$ and of $x^{x^{x}} = 10^{1,000,000}$ are +approximately $2.68$~and~$7.11$. + +\Item{(v)} If $x^{10} = 10^{y}$, then for $x = 100$, $y = 20$; and for $x = 10^{10}$, $y = 100$. + +If $x^{10^{10}} = 10^{10^{y}}$, then for $x = 100$, $y = 10.30$; for $x = 10^{10}$, $y = 11$; and +for $x = 10^{10^{10}}$, $y = 20$. + +If $x^{10^{10^{10}}} = 10^{10^{10^{y}}}$, then for $x = 10^{10}$, $y = 10 + (4.3 × 10^{-11})$; for $x = 10^{10^{10}}$, +$y = 10 + (4.3 × 10^{-10})$; and for $x = 10^{10^{10^{10}}}$, $y = 10.30$. + +\Section[7.]{Some numbers of physics.} + +The distance to $\alpha$~Centauri is roughly $26,000,000,000,000$ miles or +$1.65 × 10^{18}$~inches. The number of inches lies between $19!$~and~$20!$ and +is approximately equal to~$e^{e^{3.74}}$ or~$16^{e^{e}}$. Again, writing $15$~letters to the +inch (an average size in print) a line to the star would be sufficient +for the writing at length of~$10^{2.47 × 10^{19}}$. The latter number is approximately +equal to $(14 × 10^{17})!$, $e^{e^{e^{3.83}}}$, or $(10^{6.5 × 10^{12}})^{e^{e^{e}}}$. + +If we take the distance to the end of the visible universe to be that +through which light travels in $10,000$~years, we find that this distance +when expressed in wave-lengths of sodium light is measured roughly +by the numbers +\[ +1.6 ×10^{26},\quad 26!,\quad e^{e^{4.10}},\quad (53.6)^{e^{e}},\quad 3.29^{3.29^{3.29}}. +\] + +If we assume the average distance between the centres of two +adjacent molecules of the earth's substance to be $10^{-8}$~cm., we find +that the number of molecules in the earth is roughly +\[ +10.8 × 10^{50},\quad 42!,\quad e^{e^{4.77}},\quad (2333)^{e^{e}},\quad 3.56^{3.56^{3.56}}. +\] +\vfill +\hrule +\Strut[8pt] +{\scriptsize CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS} +%% -----File: 071.png---Folio 63------- +%% -----File: 072.png---Folio 64------- +%% -----File: 073.png---Folio 65------- +%% -----File: 074.png---Folio 66------- +\clearpage +\thispagestyle{empty} +\begin{center} +\Titlefont{Cambridge Tracts in Mathematics\\ + and Mathematical Physics}\\ +\rule{1.5in}{1.0pt} +\end{center} + +{\footnotesize +\Catalog{No.~1.} VOLUME AND SURFACE INTEGRALS USED IN +PHYSICS, by \textsc{J.~G. 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Grace}, M.A., +F.R.S. + +\Inprep AN INTRODUCTION TO THE THEORY OF ATTRACTIONS, +by \textsc{T.~J.~I'A. Bromwich}, M.A., F.R.S.\par +} +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% +\FlushRunningHeads +\vfill +\PGLicense +\begin{PGtext} +End of Project Gutenberg's Orders of Infinity, by Godfrey Harold Hardy + +*** END OF THIS PROJECT GUTENBERG EBOOK ORDERS OF INFINITY *** + +***** This file should be named 38079-pdf.pdf or 38079-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/8/0/7/38079/ + +Produced by Andrew D. 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You may copy it, give it away or % +% re-use it under the terms of the Project Gutenberg License included % +% with this eBook or online at www.gutenberg.org % +% % +% % +% Title: Orders of Infinity % +% The 'Infinitärcalcül' of Paul Du Bois-Reymond % +% % +% Author: Godfrey Harold Hardy % +% % +% Release Date: November 25, 2011 [EBook #38079] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK ORDERS OF INFINITY *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{38079} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. Required. %% +%% amssymb: Additional mathematical symbols. 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You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Orders of Infinity + The 'Infinitärcalcül' of Paul Du Bois-Reymond + +Author: Godfrey Harold Hardy + +Release Date: November 25, 2011 [EBook #38079] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK ORDERS OF INFINITY *** +\end{PGtext} +\end{minipage} +\end{center} +\clearpage + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Brenda Lewis and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive/Canadian Libraries) +\end{PGtext} +\end{minipage} +\vfill +\TranscribersNote{\TransNoteText} +\end{center} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +%% -----File: 001.png---Folio xx------- +\cleardoublepage +\pagenumbering{roman} +\null\vfill +\begin{center} +\Titlefont{Cambridge Tracts in Mathematics \\[12pt] +and Mathematical Physics} +\bigskip + +\textsc{General Editors} +\medskip + +J. G. LEATHEM, M.A. \\ +E. T. WHITTAKER, M.A., F.R.S. +\vfill + +\Titlefont{No.\ 12 \\[24pt] +ORDERS OF INFINITY} +\end{center} +%% -----File: 002.png---Folio xx------- +\clearpage +\begin{center} +\large +CAMBRIDGE UNIVERSITY PRESS \\ +\textgoth{London}: FETTER LANE, E.C. \\ +C. F. CLAY, \textsc{Manager} +\bigskip + +\Graphic[png]{1.25in}{cups} +\bigskip + +\normalsize +\textgoth{Edinburgh}: 100, PRINCES STREET \\ +\textgoth{Berlin}: A. ASHER AND CO. \\ +\textgoth{Leipzig}: F. A. BROCKHAUS \\ +\textgoth{New York}: G. P. PUTNAM'S SONS \\ +\textgoth{Bombay and Calcutta}: MACMILLAN AND CO., \textsc{Ltd.} +\vfill + +\textit{All rights reserved} +\end{center} +%% -----File: 003.png---Folio xx------- +\clearpage +\begin{center} +\Titlefont{\Huge ORDERS OF INFINITY} +\bigskip + +{\large THE `INFINITÄRCALCÜL' OF \\[8pt] +PAUL DU BOIS-REYMOND} +\vfill +\vfill + +by +\bigskip + +G. H. HARDY, M.A., F.R.S. \\ +\medskip + +{\small Fellow and Lecturer of Trinity College, Cambridge} +\vfill +\vfill +\vfill + +{\large Cambridge: \\ +at the University Press + +1910} +\end{center} +%% -----File: 004.png---Folio xx------- +\clearpage +\null\vfill +\begin{center} +\textgoth{Cambridge}: +\medskip + +\footnotesize +PRINTED BY JOHN CLAY, M.A. +\medskip + +AT THE UNIVERSITY PRESS +\end{center} +\vfill +%% -----File: 005.png---Folio xx------- + +\Preface + +\First{The} ideas of Du~Bois-Reymond's \textit{Infinitärcalcül} are of great and +growing importance in all branches of the theory of functions. +With the particular system of notation that he invented, it is, no +doubt, quite possible to dispense; but it can hardly be denied that +the notation is exceedingly useful, being clear, concise, and expressive +in a very high degree. In any case Du~Bois-Reymond was a mathematician +of such power and originality that it would be a great pity if +so much of his best work were allowed to be forgotten. + +There is, in Du~Bois-Reymond's original memoirs, a good deal that +would not be accepted as conclusive by modern analysts. He is also +at times exceedingly obscure; his work would beyond doubt have +attracted much more attention had it not been for the somewhat +repugnant garb in which he was unfortunately wont to clothe his most +valuable ideas. I have therefore attempted, in the following pages, +to bring the \textit{Infinitärcalcül} up to date, stating explicitly and proving +carefully a number of general theorems the truth of which Du~Bois-Reymond +seems to have tacitly assumed---I may instance in particular +the theorem of~\Ref{iii.}{§\;2}. + +I have to thank Messrs J.~E. Littlewood and G.~N. Watson for +their kindness in reading the proof-sheets, and Mr J.~Jackson for the +numerical results contained in Appendix~III\@. + +\Signature{G. H. H.} +{\textsc{Trinity College},} +{\textit{April}, 1910.} +%% -----File: 006.png---Folio xx------- +%% -----File: 007.png---Folio xx------- +\Contents + +\PageLine + +\ToCChap{I.}{Introduction}{1} + +\ToCChap{II.}{Scales of infinity in general}{7} + +\ToCChap{III.}{Logarithmico-exponential scales}{16} + +\ToCChap{IV.}{Special problems connected with logarithmico-exponential +scales}{21} + +\ToCChap{V.}{Functions which do not conform to any logarithmico-exponential +scale}{26} + +\ToCChap{VI.}{Differentiation and integration}{36} + +\ToCChap{VII.}{Some developments of Du Bois-Reymond's \textit{Infinitärcalcül}}{41} + +% Prints heading "Appendix I." +\ToCApp{I.}{General Bibliography}{47} + +\ToCApp{II.}{A sketch of some applications, with references}{48} + +\ToCApp{III.}{Some numerical results}{58} + +%% -----File: 008.png---Folio xx------- +%% -----File: 009.png---Folio 1------- +\MainMatter + +\Chapter{I.}{Introduction.} + +\Paragraph{1.} \First{The} notions of the `order of greatness' or `order of smallness' +of a function~$f(n)$ of a positive integral variable~$n$, when $n$~is `large,' +or of a function~$f(x)$ of a continuous variable~$x$, when $x$~is `large' or +`small' or `nearly equal to~$a$,' are of the greatest importance even in +the most elementary stages of mathematical analysis.\footnote + {See, for instance, my \textit{Course of pure mathematics}, pp.~168~\textit{et seq.}, 183~\textit{et seq.}, + 344~\textit{et seq.}, 350.} +The student +soon learns that as $x$~tends to infinity ($x \to \infty$) then also $x^{2} \to \infty$, and +moreover that $x^{2}$~tends to infinity \emph{more rapidly than~$x$}, \ie\ that the +ratio~$x^{2}/x$ tends to infinity as well; and that $x^{3}$~tends to infinity more +rapidly than~$x^{2}$, and so on indefinitely: and it is not long before he +begins to appreciate the idea of a `scale of infinity'~$(x^{n})$ formed by the +functions $x$,~$x^{2}$, $x^{3}$,~\dots, $x^{n}$,~\dots. This scale he may supplement and to +some extent complete by the interpolation of fractional powers of~$x$, +and, when he is familiar with the elements of the theory of the +logarithmic and exponential functions, of irrational powers: and so he +obtains a scale~$(x^{\alpha})$, where $\alpha$~is any positive number, formed by all +possible positive powers of~$x$. He then learns that there are functions +whose rates of increase cannot be measured by any of the functions of +this scale: that $\log x$, for example, tends to infinity more slowly, and $e^{x}$ +more rapidly, than \emph{any} power of~$x$; and that $x/(\log x)$ tends to infinity +more slowly than~$x$, but more rapidly than any power of~$x$ less than +the first. + +As we proceed further in analysis, and come into contact with its +most modern developments, such as the theory of Fourier's series, +the theory of integral functions, or the theory of singular points of +analytic functions, the importance of these ideas becomes greater and +%% -----File: 010.png---Folio 2------- +greater. It is the systematic study of them, the investigation of +general theorems concerning them and ready methods of handling +them, that is the subject of Paul du~Bois-Reymond's \textit{Infinitärcalcül} +or `calculus of infinities.' + +\Paragraph{2.} The notion of the `order' or the `rate of increase' of a function +is essentially a relative one. If we wish to say that `the rate of +increase of~$f(x)$ is so and so' all we can say is that it is greater than, +equal to, or less than that of some other function~$\phi(x)$. + +Let us suppose that $f$~and~$\phi$ are two functions of the continuous +variable~$x$, defined for all values of~$x$ greater than a given value~$x_{0}$. +Let us suppose further that $f$~and~$\phi$ are positive, continuous, and +steadily increasing functions which tend to infinity with~$x$; and let us +consider the ratio~$f/\phi$. We must distinguish four cases: + +\Item{(i)} If $f/\phi \to \infty$ with~$x$, we shall say that the rate of increase, or +simply the \emph{increase}, of~$f$ is greater than that of~$\phi$, and shall write +\[ +f \cgt \phi. +\] + +\Item{(ii)} If $f/\phi \to 0$, we shall say that the increase of~$f$ is less than that +of~$\phi$, and write +\[ +f \clt \phi. +\] + +\Item{(iii)} If $f/\phi$ remains, for all values of~$x$ however large, between two +fixed positive numbers $\delta$,~$\Delta$, so that $0 < \delta < f/\phi < \Delta$, we shall say that +the increase of~$f$ is equal to that of~$\phi$, and write +\[ +f \ceq \phi. +\] + +It may happen, in this case, that $f/\phi$ actually tends to a definite +limit. If this is so, we shall write +\[ +f \ceqq \phi. +\] + +Finally, if this limit is \emph{unity}, we shall write +\[ +f \sim \phi. +\] + +When we can compare the increase of~$f$ with that of some standard +function~$\phi$ by means of a relation of the type $f \ceq \phi$, we shall say that +$\phi$~\emph{measures}, or simply \emph{is}, the increase of~$f$. Thus we shall say that +the increase of~$2x^{2} + x + 3$ is~$x^{2}$. + +It usually happens in applications that $f/\phi$~is monotonic (\ie\ +steadily increasing or steadily decreasing) as well as $f$~and~$\phi$ themselves. +It is clear that in this case $f/\phi$ must tend to infinity, or zero, +or to a positive limit: so that one of the three cases indicated above +%% -----File: 011.png---Folio 3------- +must occur, and we must have $f \cgt \phi$ or $f \clt \phi$ or $f \ceqq \phi$ (not merely +$f \ceq \phi$). We shall see in a moment that this is not true in general. + +\Item{(iv)} It may happen that $f/\phi$ neither tends to infinity nor to zero, +nor remains between fixed positive limits. + +\begin{Remark} +Suppose, for example, that $\phi_{1}$,~$\phi_{2}$ are two continuous and increasing +functions such that $\phi_{1} \cgt \phi_{2}$. A glance at the +figure (\Fig{1}) will probably show with sufficient +% [Illustration: Fig. 1.] +\Figure[0.45\textwidth]{1}{011} +clearness how we can construct, by means of a +`staircase' of straight or curved lines, running +backwards and forwards between the graphs of +$\phi_{1}$~and~$\phi_{2}$, the graph of a steadily increasing +function~$f$ such that $f = \phi_{1}$ for $x = x_{1}$, $x_{3}$,~\dots\ and +$f = \phi_{2}$ for $x = x_{2}$, $x_{4}$,~\dots. Then $f/\phi_{1} = 1$ for +%[** TN: Next line displayed in the original] +$x = x_{1}$, $x_{3}$,~\dots, +but assumes for $x = x_{2}$, $x_{4}$,~\dots\ values which +decrease beyond all limit; while $f/\phi_{2} = 1$ +for $x = x_{2}$, $x_{4}$,~\dots, but assumes for $x = x_{1}$, $x_{3}$,~\dots\ +values which increase beyond all limit; and $f/\phi$, +where $\phi$~is a function such that $\phi_{1} \cgt \phi \cgt \phi_{2}$, +as \eg\ $\phi = \sqrt{\phi_{1} \phi_{2}}$, assumes both values which +increase beyond all limit and values which +decrease beyond all limit. + +Later on (\Ref{v.}{§\;3}) we shall meet with cases of this kind in which the +functions are defined by explicit analytical formulae. +\end{Remark} + +\Paragraph{3.} If a positive constant~$\delta$ can be found such that $f > \delta \phi$ for all +sufficiently large values of~$x$, we shall write +\[ +f \cgeq \phi; +\] +and if a positive constant~$\Delta$ can be found such that $f < \Delta \phi$ for all +sufficiently large values of~$x$, we shall write +\[ +f \cleq \phi. +\] +If $f \cgeq \phi$ and $f \cleq \phi$, then $f \ceq \phi$. + +It is however important to observe (i)~that $f \cgeq \phi$ is not logically +equivalent to the negation of $f \clt \phi$\footnote + {The relations $f \cgeq \phi$, $f \clt \phi$ are mutually exclusive but not exhaustive: $f \cgeq \phi$ + implies the negation of $f \clt \phi$, but the converse is not true.} +and (ii)~that it is not logically +equivalent to the alternative `\emph{$f \cgt \phi$ or $f \ceq \phi$}.' Thus, in the example +discussed at the end of~§\;2, $\phi_{1} \cgeq f \cgeq \phi_{2}$, but no one of the relations +$\phi_{1} \cgt f$, etc.\ holds. If however we know that one of the relations +$f \cgt \phi$, $f \ceq \phi$, $f \clt \phi$ \emph{must} hold, then these various assertions \emph{are} +logically equivalent. +%% -----File: 012.png---Folio 4------- + +The reader will be able to prove without difficulty that the symbols +$\cgt$,~$\ceq$,~$\clt$ satisfy the following theorems. +\begin{align*} +&\text{If $f \cgt \phi$, $\phi \cgeq \psi$, then $f \cgt \psi$.} \\ +&\text{If $f \cgeq \phi$, $\phi \cgt \psi$, then $f \cgt \psi$.} \\ +&\text{If $f \cgeq \phi$, $\phi \cgeq \psi$, then $f \cgeq \psi$.} \\ +&\text{If $f \ceq \phi$, $\phi \ceq \psi$, then $f \ceq \psi$.} +\displaybreak[1] \\[6pt] +&\text{If $f \cgeq \phi$, then $f + \phi \ceq f$.} \\ +&\text{If $f \cgt \phi$, then $f - \phi \ceq f$.} +\displaybreak[1] \\[6pt] +&\text{If $f \cgt \phi$, $f_{1} \cgt \phi_{1}$, then $f + f_{1} \cgt \phi + \phi_{1}$.} \\ +&\text{If $f \cgt \phi$, $f_{1} \ceq \phi_{1}$, then $f + f_{1} \cgeq \phi + \phi_{1}$.} \\ +&\text{If $f \ceq \phi$, $f_{1} \ceq \phi_{1}$, then $f + f_{1} \ceq \phi + \phi_{1}$.} +\displaybreak[1] \\[6pt] +&\text{If $f \cgt \phi$, $f_{1} \cgeq \phi_{1}$, then $ff_{1} \cgt \phi \phi_{1}$.} \\ +&\text{If $f \ceq \phi$, $f_{1} \ceq \phi_{1}$, then $ff_{1} \ceq \phi \phi_{1}$.} +\end{align*} + +Many other obvious results of the same character might be stated, +but these seem the most important. The reader will find it instructive +to state for himself a series of similar theorems involving also the +symbols $\ceqq$~and~$\sim$. + +\Paragraph{4.} So far we have supposed that the functions considered all tend +to infinity with~$x$. There is nothing to prevent us from including also +the case in which $f$~or~$\phi$ tends steadily to zero, or to a limit other than +zero. Thus we may write $x \cgt 1$, or $x \cgt 1/x$, or $1/x \cgt 1/x^{2}$. Bearing +this in mind the reader should frame a series of theorems similar to +those of~§\;3 but having reference to \emph{quotients} instead of to sums or +products. + +It is also convenient to extend our definitions so as to apply to +\emph{negative} functions which tend steadily to~$-\infty$ or to~$0$ or to some other +limit. In such cases we make no distinction, when using the symbols +$\cgt$,~$\clt$, $\ceq$,~$\ceqq$, between the function and its modulus: thus we write +$-x \clt -x^{2}$ or $-1/x \clt 1$, meaning thereby exactly the same as by +$x \clt x^{2}$ or $1/x \clt 1$. But $f \sim \phi$ is of course to be interpreted as a +statement about the actual functions and not about their moduli. + +It will be well to state at this point, once for all, that all functions +referred to in this tract, from here onwards, are to be understood, +unless the contrary is expressly stated or obviously implied, to be +positive, continuous, and monotonic, increasing of course if they tend +to~$\infty$, and decreasing if they tend to~$0$. But it is sometimes convenient +%% -----File: 013.png---Folio 5------- +to use our symbols even when this is not true of all the +functions concerned; to write, for example, +\[ +1 + \sin x \clt x, \qquad +x^{2} \cgt x\sin x, +\] +meaning by the first formula simply that $|1 + \sin x|/x \to 0$. This +kind of use may clearly be extended even to complex functions +(\eg~$e^{ix} \clt x$). + +Again, we have so far confined our attention to functions of a +continuous variable~$x$ which tends to~$+\infty$. This case includes that +which is perhaps even more important in applications, that of functions +of the positive integral variable~$n$: we have only to disregard values of~$x$ +other than integral values. Thus $n! \cgt n^{2}$, $-1/n \clt n$. + +Finally, by putting $x = -y$, $x = 1/y$, or $x = 1/(y - a)$, we are led to +consider functions of a continuous variable~$y$ which tends to~$-\infty$ or~$0$ +or~$a$: the reader will find no difficulty in extending the considerations +which precede to cases such as these. + +In what follows we shall generally state and prove our theorems +only for the case with which we started, that of indefinitely increasing +functions of an indefinitely increasing continuous variable, and shall +leave to the reader the task of formulating the corresponding theorems +for the other cases. We shall in fact always adopt this course, except +on the rare occasions when there is some essential difference between +different cases. + +\Paragraph{5.} There are some other symbols which we shall sometimes find it +convenient to use in special senses. + +By +\[ +O(\phi) +\] +we shall denote a function~$f$, otherwise unspecified, but such that +\[ +|f| < K\phi, +\] +where $K$~is a positive constant, and $\phi$~a positive function of~$x$: this +notation is due to Landau. Thus +\[ +x + 1 = O(x), \qquad +x = O(x^{2}), \qquad +\sin x = O(1). +\] + +We shall follow Borel in using the same letter~$K$ in a whole series +of inequalities to denote a positive constant, not necessarily the same +in all inequalities where it occurs. Thus +\[ +\sin x < K, \qquad +2x + 1 < Kx, \qquad +x^{m} < Ke^{x}. +\] +{\Loosen If we use~$K$ thus in any finite number of inequalities which (like the +first two above) do not involve any variables other than~$x$, or whatever +other variable we are primarily considering, then all the values of~$K$ lie +%% -----File: 014.png---Folio 6------- +between certain absolutely fixed limits $K_{1}$~and~$K_{2}$ (thus $K_{1}$~might be +$10^{-10}$ and $K_{2}$~be~$10^{10}$). In this case all the~$K$'s satisfy $0 < K_{1} < K < K_{2}$, +and every relation $f < K\phi$ might be replaced by $f < K_{2}\phi$, and every +relation $f > K\phi$ by $f > K_{1}\phi$. But we shall also have occasion to use $K$ +in equalities which (like the third above) involve a parameter (here~$m$). +In this case $K$, though independent of~$x$, is a function of~$m$. Suppose +that $\alpha$,~$\beta$,~\dots\ are all the parameters which occur in this way in this +tract. Then if we give any special system of values to $\alpha$,~$\beta$,~\dots, we +can determine $K_{1}$,~$K_{2}$ as above. Thus all our $K$'s satisfy} +\[ +0 < K_{1}(\alpha, \beta, \dots) < K < K_{2}(\alpha, \beta, \dots), +\] +where $K_{1}$,~$K_{2}$ are positive functions of $\alpha$,~$\beta$,~\dots\ defined for any permissible +set of values of those parameters. But $K_{1}$~has zero for its +lower limit; by choosing $\alpha$,~$\beta$,~\dots\ appropriately we can make~$K_{1}$ as +small as we please---and, of course, $K_{2}$~as large as we please.\footnote + {I am indebted to Mr~Littlewood for the substance of these remarks.} + +It is clear that the three assertions +\[ +f = O(\phi), \qquad +|f| < K\phi, \qquad +f \cleq \phi +\] +are precisely equivalent to one another. + +When a function~$f$ possesses any property for all values of~$x$ greater +than some definite value (this value of course depending on the nature +of the particular property) we shall say that $f$~possesses the property +for $x > x_{0}$. Thus +\[ +x > 100 \quad (x > x_{0}), \qquad +e^{x} > 100 x^{2} \quad (x > x_{0}). +\] + +We shall use $\delta$ to denote an arbitrarily small but fixed positive +number, and $\Delta$~to denote an arbitrarily great but likewise fixed positive +number. Thus +\[ +f < \delta\phi \quad (x > x_{0}) +\] +means `however small~$\delta$, we can find~$x_{0}$ so that $f < \delta\phi$ for $x > x_{0}$,' +\ie\ means the same as $f \clt \phi$; and $\phi > \Delta f\ (x > x_{0})$ means the same: +and +\[ +(\log x)^{\Delta} \clt x^{\delta} +\] +means `any power of~$\log x$, however great, tends to infinity more +slowly than any positive power of~$x$, however small.' + +Finally, we denote by~$\epsilon$ a function (of a variable or variables +indicated by the context or by a suffix) whose limit is zero when the +variable or variables are made to tend to infinity or to their limits +in the way we happen to be considering. Thus +\[ +f = \phi(1 + \epsilon), \qquad +f \sim \phi +\] +are equivalent to one another. +%% -----File: 015.png---Folio 7------- + +In order to become familiar with the use of the symbols defined in the +preceding sections the reader is advised to verify the following relations; in +them $P_{m}(x)$,~$Q_{n}(x)$ denote polynomials whose degrees are $m$~and~$n$ and whose +leading coefficients are positive: +\begin{gather*} +P_{m}(x) \cgt Q_{n}(x) \quad (m > n), \qquad + P_{m}(x) \ceqq Q_{n}(x) \quad (m = n), \\ +P_{m}(x) \ceqq x^{m}, \qquad + P_{m}(x)/Q_{n}(x) \ceqq x^{m-n}, +\displaybreak[1] \\[6pt] +\sqrt{ax^{2} + 2bx + c} \ceqq x \quad (a > 0), \qquad + \sqrt{x + a} \sim \sqrt{x}, \\ +\sqrt{x + a} - \sqrt{x} \sim a/2\sqrt{x}, \qquad + \sqrt{x + a} - \sqrt{x} = O(1/\sqrt{x}), +\displaybreak[1] \\[6pt] +e^{x} \cgt x^{\Delta}, \qquad + e^{x^{2}} \cgt e^{\Delta x}, \qquad + e^{e^{x}} \cgt e^{x^{\Delta}}, \\ +\log x \clt x^{\delta}, \quad + \log P_{m}(x) \ceqq \log Q_{n}(x), \quad + \log \log P_{m}(x) \sim \log \log Q_{n}(x), +\displaybreak[1] \\[6pt] +x + a\sin x \sim x, \qquad + x(a + \sin x) \ceq x\quad (a > 1), \\ +e^{a + \sin x} \ceq 1, \qquad + \cosh x \sim \sinh x \ceqq e^{x}, \\ +x^{m} = O(e^{\delta x}), \qquad + (\log x)/x = O(x^{\delta-1}), +\displaybreak[1] \\[6pt] +1 + \frac{1}{2} + \dots + \frac{1}{n} \cgt 1, \qquad + 1 + \frac{1}{2^{2}} + \dots + \frac{1}{n^{2}} \ceqq 1, \\ +1 + \frac{1}{2} + \dots + \frac{1}{n} \sim \log n, \qquad + 1 + \frac{1}{2} + \dots + \frac{1}{n} - \log n \ceqq 1, \\ +n! \clt n^{n}, \qquad + n! \cgt e^{\Delta n}, \qquad + n! = n^{n^{1+\epsilon}} = n^{n(1 + \epsilon)}, \\ +n! \sim n^{n + \frac{1}{2}} e^{-n} \sqrt{2\pi}, \qquad + n!\, (e/n)^{n} = (1 + \epsilon) \sqrt{2\pi n}, \\ +\int_{1}^{x} \frac{dt}{t} \cgt 1, \qquad + \int_{1}^{x} \frac{dt}{t} \sim \log x, \qquad + \int_{2}^{x} \frac{dt}{\log t} \sim \frac{x}{\log x}. +\end{gather*} + + +\Chapter{II.}{Scales of infinity in general.} + +\Paragraph{1.} \First{If} we start from a function~$\phi$, such that $\phi \cgt 1$, we can, in a +variety of ways, form a series of functions +\[ +\phi_{1} = \phi,\quad +\phi_{2},\quad +\phi_{3},\ \dots,\quad +\phi_{n},\ \dots +\] +such that the increase of each function is greater than that of its +predecessor. Such a sequence of functions we shall denote for shortness +by~$(\phi_{n})$. + +One obvious method is to take $\phi_{n} = \phi^{n}$. Another is as follows: +If $\phi \cgt x$, it is clear that +\[ +\phi\{\phi(x)\} / \phi(x) \to \infty, +\] +%% -----File: 016.png---Folio 8------- +and so $\phi_{2}(x) = \phi \phi(x) \cgt \phi(x)$; similarly $\phi_{3}(x) = \phi \phi_{2}(x) \cgt \phi_{2}(x)$, and +so on.\footnote + {For some results as to the increase of such iterated functions see \Ref{vii.}{§\;2~(vi)}.} + +Thus the first method, with $\phi = x$, gives the scale $x$,~$x^{2}$, $x^{3}$,~\dots\ or~$(x^{n})$; +the second, with $\phi = x^{2}$, gives the scale $x^{2}$,~$x^{4}$, $x^{8}$,~\dots\ or~$(x^{2^{n}})$. + +\begin{Remark} +These scales are \emph{enumerable} scales, formed by a simple progression of +functions. We can also, of course, by replacing the integral parameter~$n$ by +a continuous parameter~$\alpha$, define scales containing a non-enumerable +multiplicity of functions: the simplest is~$(x^{\alpha})$, where $\alpha$~is any positive number. +But such scales fill a subordinate \textit{rôle} in the theory. +\end{Remark} + +It is obvious that we can always insert a new term (and therefore, +of course, any number of new terms) in a scale at the beginning or +between any two terms: thus $\sqrt{\phi}$ (or $\phi^{\alpha}$, where $\alpha$~is any positive +number less than unity) has an increase less than that of any term +of the scale, and $\sqrt{\phi_{n} \phi_{n+1}}$ or $\phi_{n}^{\alpha} \phi_{n+1}^{1-\alpha}$ has an increase intermediate +between those of $\phi_{n}$~and~$\phi_{n+1}$. A less obvious and far more important +theorem is the following + +\begin{Result}[Theorem of Paul du~Bois-Reymond.] Given any ascending +scale of increasing functions~$\phi_{n}$, \ie\ a series of functions such that +$\phi_{1} \clt \phi_{2} \clt \phi_{3} \clt \dots$, we can always find a function~$f$ which increases +more rapidly than any function of the scale, \ie\ which satisfies the +relation $\phi_{n} \clt f$ for all values of~$n$. +\end{Result} + +In view of the fundamental importance of this theorem we shall +give two entirely different proofs. + +\Paragraph{2.} (i)~We know that $\phi_{n+1} \cgt \phi_{n}$ for all values of~$n$, but this, of +course, does not necessarily imply that $\phi_{n+1} \geq \phi_{n}$ for all values of $x$~and~$n$ +in question.\footnote + {$\phi_{n+1} \cgt \phi_{n}$ implies $\phi_{n+1} > \phi_{n}$ for sufficiently large values of~$x$, say for $x > x_{n}$. + But $x_{n}$ may tend to~$\infty$ with~$n$. Thus if $\phi_{n} = x^{n}/n!$ we have $x_{n} = n + 1$.} +We can, however, construct a new scale of +functions~$\psi_{n}$ such that + +\Item{(\textit{a})} $\psi_{n}$ is identical with~$\phi_{n}$ for all values of~$x$ from a certain value +$x_{n}$ onwards ($x_{n}$, of course, depending upon~$n$); + +\Item{(\textit{b})} $\psi_{n+1} \geq \psi_{n}$ for all values of $x$~and~$n$. + +For suppose that we have constructed such a scale up to its $n$th~term~$\psi_{n}$. +Then it is easy to see how to construct~$\psi_{n+1}$. Since +$\phi_{n+1} \cgt \phi_{n}$, $\phi_{n} \sim \psi_{n}$, it follows that $\phi_{n+1} \cgt \psi_{n}$, and so $\phi_{n+1} > \psi_{n}$ from a +certain value of~$x$ (say~$x_{n+1}$) onwards. For $x \geq x_{n+1}$ we take $\psi_{n+1} = \phi_{n+1}$. +For $x < x_{n+1}$ we give $\psi_{n+1}$ a value equal to the greater of the values of +%% -----File: 017.png---Folio 9------- +$\phi_{n+1}$,~$\psi_{n}$. Then it is obvious that $\psi_{n+1}$~satisfies the conditions (\textit{a})~and~(\textit{b}). + +Now let +\[ +f(n) = \psi_{n}(n). +\] +From $f(n)$ we can deduce a continuous and increasing function~$f(x)$, +such that +\[ +\psi_{n}(x) < f(x) < \psi_{n+1}(x) +\] +for $n < x < n + 1$, by joining the points~$(n, \psi_{n}(n))$ by straight lines or +suitably chosen arcs of curves. + +\begin{Remark} +It is perhaps worth while to call attention explicitly to a small point that +has sometimes been overlooked (see, \eg, +Borel, \textit{Leçons sur la théorie des fonctions}, +p.~114; \textit{Leçons sur les séries à termes positifs}, +p.~26). It is not always the case that the +use of straight lines will ensure +\[ +f(x) > \psi_{n}(x) +\] +for $x > n$ (see, for example, \Fig{2}, where +the dotted line represents an appropriate +arc). +\end{Remark} +% [Illustration: Fig. 2.] +\Figure{2}{017} + +Then +\[ +f/\psi_{n} > \psi_{n+1}/\psi_{n} +\] +for $x > n + 1$, and so $f \cgt \psi_{n}$; therefore +$f \cgt \phi_{n}$ and the theorem is proved. + +\begin{Remark} +{\Loosen The proof which precedes may be made +more general by taking $f(n) = \psi_{\lambda_{n}} (n)$, where +$\lambda_{n}$~is an integer depending upon~$n$ and +tending steadily to infinity with~$n$.} +\end{Remark} + +(ii)~The second proof of Du~Bois-Reymond's Theorem proceeds on +entirely different lines. We can always choose positive coefficients~$a_{n}$ +so that +\[ +f(x) = \sum_{1}^{\infty} a_{n}\psi_{n}(x) +\] +is convergent for all values of~$x$. This will certainly be the case, for +instance, if +\[ +1/a_{n} = \psi_{1}(1) \psi_{2}(2) \dots \psi_{n}(n). +\] +For then, if $\nu$~is any integer greater than~$x$, $\psi_{n}(x) < \psi_{n}(n)$ for $n \geqq \nu$, +and the series will certainly be convergent if +\[ +\sum_{\nu}^{\infty} \frac{1}{\psi_{1}(1) \psi_{2}(2) \dots \psi_{n-1}(n-1)} +\] +is convergent, as is obviously the case. +%% -----File: 018.png---Folio 10------- + +Also +\[ +f(x)/\psi_{n}(x) > a_{n+1}\psi_{n+1}(x)/\psi_{n}(x) \to \infty, +\] +so that $f \cgt \phi_{n}$ for all values of~$n$. + +\begin{Remark} +\Paragraph{3.} Suppose, \eg, that $\phi_{n} = x^{n}$. If we restrict ourselves to values of~$x$ +greater than~$1$, we may take $\psi_{n} = \phi_{n} = x^{n}$. The first method of construction +would naturally lead to +\[ +f = n^{n} = e^{n\log n}, +\] +or $f = (\lambda_{n})^{n}$, where $\lambda_{n}$~is defined as at the end of §\;2~(i), and each of these functions +has an increase greater than that of any power of~$n$. The second method +gives +\[ +f(x) = \sum_{1}^{\infty} \frac{x^{n}}{1^{1} 2^{2} 3^{3} \dots n^{n}}. +\] + +It is known\footnote + {\textit{Messenger of Mathematics,} vol.~34, p.~101.} +that when $x$~is large the order of magnitude of this function +is roughly the same as that of +\[ +e^{\frac{1}{2}(\log x)^{2}/\log\log x}. +\] + +{\Loosen As a matter of fact it is by no means necessary, in general, in order to +ensure the convergence of the series by which $f(x)$~is defined, to suppose that +$a_{n}$~decreases so rapidly. It is very generally sufficient to suppose $1/a_{n} = \phi_{n}(n)$: +this is always the case, for example, if $\phi_{n}(x) = \{\phi(x)\}^{n}$, as the series} +\[ +\sum \left\{\frac{\phi(x)}{\phi(n)}\right\}^{n} +\] +is always convergent. This choice of~$a_{n}$ would, when $\phi = x$, lead to +\[ +f(x) = \sum \left(\frac{x}{n}\right)^{n} + \sim \sqrt{\frac{2\pi x}{e}}\, e^{x/e}.\footnote + {Lindelöf, \textit{Acta Societatis Fennicae}, t.~31, p.~41; Le~Roy, \textit{Bulletin des Sciences + Mathématiques}, t.~24, p.~245.\PageLabel{10}} +\] + +But the simplest choice here is $1/a_{n} = n!$, when +\[ +f(x) = \sum \frac{x^{n}}{n!} = e^{x} - 1; +\] +it is naturally convenient to disregard the irrelevant term~$-1$. + +\Paragraph{4.} We can always suppose, if we please, that $f(x)$~is defined by a power +series $\sum a_{n}x^{n}$ convergent for all values of~$x$, in virtue of a theorem of Poincaré's\footnote + {\textit{American Journal of Mathematics}, vol.~14, p.~214.} +which is of sufficient intrinsic interest to deserve a formal statement and +proof. + +\begin{Result} +Given any continuous increasing function~$\phi(x)$, we can always find an +integral function~$f(x)$ \(\ie\ a function~$f(x)$ defined by a power series $\sum a_{n}x^{n}$ +convergent for all values of~$x$\) such that $f(x) \cgt \phi(x)$. +\end{Result} + +The following simple proof is due to Borel.\footnote + {\textit{Leçons sur les séries à termes positifs}, p.~27.} + +Let $\Phi(x)$ be any function (such as the square of~$\phi$) such that $\Phi \cgt \phi$. Take +%% -----File: 019.png---Folio 11------- +an increasing sequence of numbers~$a_{n}$ such that $a_{n} \to \infty$, and another sequence +of numbers~$b_{n}$ such that +\[ +a_{1} < b_{2} < a_{2} < b_{3} < a_{3} < \dots; +\] +and let +\[ +f(x) = \sum \left(\frac{x}{b_{n}}\right)^{\nu_{n}}, +\] +where $\nu_{n}$~is an integer and $\nu_{n+1} > \nu_{n}$. This series is convergent for all values +of~$x$; for the $n$th~root of the $n$th~term is, for sufficiently large values of~$n$, not +greater than~$x/b_{n}$, and so tends to zero. Now suppose $a_{n} \leqq x < a_{n+1}$; then +\[ +f(x) > \left(\frac{a_{n}}{b_{n}}\right)^{\nu_{n}}. +\] +Since $a_{n} > b_{n}$ we can suppose $\nu_{n}$~so chosen that (i)~$\nu_{n}$~is greater than any of +$\nu_{1}$,~$\nu_{2}$, \dots,~$\nu_{n-1}$ and (ii) +\[ +\left(\frac{a_{n}}{b_{n}}\right)^{\nu_{n}} > \Phi(a_{n+1}). +\] + +Then +\[ +f(x) > \Phi(a_{n+1}) > \Phi(x), +\] +and so $f \cgt \phi$. +\end{Remark} + +\Paragraph{5.} So far we have confined our attention to ascending scales, such +as $x$,~$x^{2}$, $x^{3}$,~\dots, $x^{n}$,~\dots\ or~$(x^{n})$; but it is obvious that we may consider +in a similar manner \emph{descending} scales such as $x$,~$\sqrt{x}$, $\sqrt[3]{x}$,~\dots, $\sqrt[n]{x}$,~\dots\ +or~$(\sqrt[n]{x})$. It is very generally (though not always) true that if $(\phi_{n})$~is +an ascending scale, and $\psi$~denotes the function inverse to~$\phi$, then +$(\psi_{n})$~is a descending scale. + +\begin{Remark} +If $\phi > \bar{\phi}$ for all values of~$x$ (or all values greater than some definite value), +then a glance at \Fig{3} is enough to show that if +$\psi$~and~$\bar{\psi}$ are the functions inverse to $\phi$~and~$\bar{\phi}$, +then $\psi < \bar{\psi}$ for all values of~$x$ (or all values +greater than some definite value). We have only +to remember that the graph of~$\psi$ may be obtained +from that of~$\phi$ by looking at the latter from a +different point of view (interchanging the \textit{rôles} of +$x$~and~$y$). But it is not true that $\phi \cgt \bar{\phi}$ involves +$\psi \clt \bar{\psi}$. Thus $e^{x} \cgt e^{x}/x$. The function inverse +to~$e^{x}$ is~$\log x$: the function inverse to~$e^{x}/x$ is +obtained by solving the equation $x = e^{y}/y$ with +respect to~$y$. This equation gives +\[ +y = \log x + \log y, +\] +and it is easy to see that $y \sim \log x$. +\end{Remark} +%[Illustration: Fig. 3.] +\Figure[0.4\textwidth]{3}{019} + +\begin{Result} +Given a scale of increasing functions~$\phi_{n}$ such that +\[ +\phi_{1} \cgt \phi_{2} \cgt \phi_{3} \cgt \dots \cgt 1, +\] +%% -----File: 020.png---Folio 12------- +we can find an increasing function~$f$ such that $\phi_{n} \cgt f \cgt 1$ for all values +of~$n$.\end{Result} The reader will find no difficulty in modifying the argument +of §\;2~(i) so as to establish this proposition. + +\Paragraph{6.} The following extensions of Du~Bois-Reymond's Theorem +(and the corresponding theorem for descending scales) are due to +Hadamard.\footnote + {\textit{Acta Mathematica}, t.~18, pp.~319 \textit{et seq.}} + +\begin{Result} +Given +\[ +\phi_{1} \clt \phi_{2} \clt \phi_{3} \clt \dots \clt \phi_{n} \clt \dots \clt \Phi, +\] +we can find $f$ so that $\phi_{n} \clt f \clt \Phi$ for all values of~$n$. +\end{Result} + +\begin{Result} +Given +\[ +\psi_{1} \cgt \psi_{2} \cgt \psi_{3} \cgt \dots \cgt \psi_{n} \cgt \dots \cgt \Psi, +\] +we can find $f$ so that $\psi_{n} \cgt f \cgt \Psi$ for all values of~$n$. +\end{Result} + +\begin{Result} +Given an ascending sequence~$(\phi_{n})$ and a descending sequence~$(\psi_{p})$ +such that $\phi_{n} \clt \psi_{p}$ for all values of $n$~and~$p$, we can find $f$ so that +\[ +\phi_{n} \clt f \clt \psi_{p} +\] +for all values of $n$~and~$p$. +\end{Result} + +To prove the first of these theorems we have only to observe that +\[ +\Phi/\phi_{1} \cgt \Phi/\phi_{2} \cgt \dots \cgt \Phi/\phi_{n} \cgt \dots \cgt 1, +\] +and to construct a function~$F$ (as we can in virtue of the theorem +of~§\;5) which tends to infinity more slowly than any of the functions~$\Phi/\phi_{n}$. +Then +\[ +f = \Phi/F +\] +is a function such as is required. Similarly for the second theorem. +The third is rather more difficult to prove. + +\begin{Remark} +In the first place, we may suppose that $\phi_{n+1} > \phi_{n}$ for all values of $x$~and~$n$: +for if this is not so we can modify the +definitions of the functions~$\phi_{n}$ as in §\;2~(i). +Similarly we may suppose $\psi_{p+1} < \psi_{p}$ for all +values of $x$~and~$p$. + +Secondly, we may suppose that, if $x$~is +fixed, $\phi_{n} \to \infty$ as $n \to \infty$, and $\psi_{p} \to 0$ as +$p \to \infty$. For if this is not true of the +functions given, we can replace them by +$H_{n}\phi_{n}$ and $K_{p}\psi_{p}$, where $(H_{n})$~is an increasing +sequence of constants, tending to~$\infty$ with~$n$, +and $(K_{p})$~a decreasing sequence of constants +whose limit as $p \to \infty$ is zero. +% [Illustration: Fig. 4.] +\Figure{4}{020} + +Since $\psi_{p} \cgt \phi_{n}$ but, for any given~$x$, $\psi_{p} < \phi_{n}$ +for sufficiently large values of~$n$, it is clear +(see \Fig{4}) that the curve $y = \psi_{p}$ intersects the curve $y = \phi_{n}$ for all sufficiently +large values of~$n$ (say for $n \geq n_{p}$). +%% -----File: 021.png---Folio 13------- + +At this point we shall, in order to avoid unessential detail, introduce a +restrictive hypothesis which can be avoided by a slight modification of the +argument,\footnote + {See Hadamard's original paper quoted above.} +but which does not seriously impair the generality of the result. +We shall assume that no curve $y = \psi_{p}$ intersects any curve $y = \phi_{n}$ in more +than one point; let us denote this point, if it exists, by~$P_{n, p}$. + +If $p$ is fixed, $P_{n, p}$~exists for $n > n_{p}$; similarly, if $n$~is fixed, $P_{n, p}$~exists +for $p > p_{n}$. And as either $n$~or~$p$ increases, so do both the ordinate or the +abscissa of~$P_{n, p}$. The curve~$\psi_{p}$ contains all the points~$P_{n, p}$ for which $p$~has +a fixed value: and $y = \phi_{n}$ contains all the points for which $n$~has a fixed value. + +It is clear that, in order to define a function~$f$ which tends to infinity +more rapidly than any~$\phi_{n}$ and less rapidly than any~$\psi_{p}$, all that we have to +do is to draw a curve, making everywhere a positive acute angle with each of +the axes of coordinates, and crossing all the curves $y = \phi_{n}$ from below to +above, and all the curves $y = \psi_{p}$ from above to below. + +Choose a positive integer~$N_{p}$, corresponding to each value of~$p$, such that +(i)~$N_{p} > n_{p}$ and (ii)~$N_{p} \to \infty$ as $p \to \infty$. Then $P_{N_{p}, p}$~exists for each value of~$p$. +And it is clear that we have only to join the points $P_{N_{1}, 1}$,~$P_{N_{2}, 2}$, $P_{N_{3}, 3}$,~\dots\ by +straight lines or other suitably chosen arcs of curves in order to obtain a +curve which fulfils our purpose. The theorem is therefore established. +\end{Remark} + +\Paragraph{7.} Some very interesting considerations relating to scales of +infinity have been developed by Pincherle.\PageLabel{13}\footnote + {\textit{Memorie della Accademia delle Scienze di Bologna} (ser.~4, t.~5, p.~739).} + +We have defined $f \cgt \phi$ to mean $f/\phi \to \infty$, or, what is the same +thing, +\[ +\log f - \log \phi \to \infty. +\Tag{(1)} +\] + +We might equally well have defined $f \cgt \phi$ to mean +\[ +F(f) - F(\phi) \to \infty, +\Tag{(2)} +\] +where $F(x)$~is any function which tends steadily to infinity with~$x$ +(\eg~$x$,~$e^{x}$). Let us say that if \Eq{(2)}~holds then +\[ +f \cgt \phi \quad (F), +\Tag{(3)} +\] +so that $f \cgt \phi$ is equivalent to $f \cgt \phi\ (\log x)$. Similarly we define +$f \clt \phi\ (F)$ to mean that $F(f) - F(\phi) \to -\infty$, and $f \ceq \phi\ (F)$ to +mean that $F(f) - F(\phi)$ remains between certain fixed limits. Thus +\begin{gather*} +x + \log x \ceq x, \qquad x + \log x \cgt x \quad (x), \\ +x + 1 \ceq x\quad (x), \qquad x + 1 \cgt x \quad (e^{x}), +\end{gather*} +since $e^{x+1} - e^{x} = (e - 1)e^{x} \to \infty$. +%% -----File: 022.png---Folio 14------- + +It is clear that the more rapid the increase of~$F$, the more likely +is it to discriminate between the rates of increase of two given +functions $f$~and~$\phi$. More precisely, \begin{Result}if +\[ +f \cgt \phi \quad (F), +\] +and $\bar{F} = FF_{1}$, where $F_{1}$~is any increasing function, then will +\[ +f \cgt \phi \quad (\bar{F}). +\] +\end{Result} + +For +\[ +\bar{F}(f) - \bar{F}(\phi) = F(f) F_{1}(f) - F(\phi) F_{1}(\phi) + > \{F(f) - F(\phi)\} F_{1}(\phi) \to \infty. +\] + +\Paragraph{8.} The substance of the following theorems is due in part to +Pincherle and in part to Du Bois-Reymond.\footnote + {Pincherle, \lc; Du~Bois-Reymond, \textit{Math.\ Annalen}, Bd.~8, S.~390 \textit{et seq.}} + +\begin{Result} +\Item{1.} However rapid the increase of~$f$, as compared with that of~$\phi$, +we can so choose~$F$ that $f \ceq \phi\ (F)$. +\end{Result} + +\begin{Result} +\Item{2.} {\Loosen If $f - \phi$ is positive for $x > x_{0}$, we can so choose~$F$ that +$f \cgt \phi\ (F)$.} +\end{Result} + +\begin{Result} +{\Loosen \Item{3.} If $f - \phi$ is monotonic and not negative for $x > x_{0}$, and +$f \ceq \phi\ (F)$, however great be the increase of~$F$, then $f = \phi$ from a +certain value of~$x$ onwards.} +\end{Result} + +\Item{(1)} If $f \cgt \phi$, we may regard~$f$ as an increasing function of~$\phi$, say +\[ +f = \theta(\phi), +\] +where $\theta(x) \cgt x$. We can choose a constant~$g$ greater than~$1$, and then +choose~$X$ so that $\theta(x) > gx$ for $x > X$. Let $a$~be any number greater +than~$X$, and let +\[ +a_{1} = \theta(a), \qquad +a_{2} = \theta(a_{1}), \qquad +a_{3} = \theta(a_{2}),\ \dots. +\] +Then $(a_{n})$~is an increasing sequence, and $a_{n} \to \infty$, since $a_{n} > g^{n}a$. + +We can now construct an increasing function~$F$ such that +\[ +F(a_{n}) = \tfrac{1}{2} nK, +\] +where $K$~is a constant. Then if $a_{\nu-1} \leqq x \leqq a_{\nu}$, $a_{\nu} \leqq \theta(x) \leqq a_{\nu+1}$, and +\[ +F\{\theta(x)\} - F(x) < F(a_{\nu+1}) - F(a_{\nu-1}) < K. +\] +Accordingly $F(f) - F(\phi)$ remains less than a constant, and so the +first theorem is established. + +\Item{(2)} Let $f - \phi = \lambda$, so that $\lambda > 0$. If $\lambda$, as $x$~increases, remains +greater than a constant~$K$, then +\[ +e^{f} - e^{\phi} > (e^{K} - 1)e^{\phi} \to \infty, +\] +so that we may take $F(x) = e^{x}$. +%% -----File: 023.png---Folio 15------- + +If it is not true that $\lambda \geqq K$, $\lambda$~assumes values less than any +assignable positive number, as $x \to \infty$. Let $\bar{\lambda}(x)$ be defined as the +lower limit of~$\lambda(\xi)$ for $\xi \leqq x$. Then $\bar{\lambda}$~tends steadily to zero as $x \to \infty$, +and $\bar{\lambda} \leqq \lambda$. We may also regard $\bar{\lambda}$ as a steadily decreasing function +of~$\phi$, say $\bar{\lambda} = \mu(\phi)$. + +Let $\varpi(\phi)$ be an increasing function of~$\phi$ such that $\varpi \cgt 1/\mu$, $\mu\varpi \cgt 1$. +Then if +\begin{gather*} +F = \int^{\phi} \varpi(t)\, dt,\\ +F(f) - F(\phi) = \int_{\phi}^{\phi + \lambda} \varpi\, dt + \geqq \int_{\phi}^{\phi + \mu(\phi)} \varpi\, dt + > \mu(\phi)\varpi(\phi) \cgt 1, +\end{gather*} +and $F(x)$~fulfils the requirement of theorem~2. The third theorem is +obviously a mere corollary of the second. + +\begin{Remark} +The reader will find it instructive to deduce or prove independently the +following three theorems, which are closely analogous to those which have +just been proved. + +\begin{Result} +\Item{1.} However great be the increase of~$f$ as compared with that of~$\phi$, we can +determine an increasing function~$F$ such that $F(f) \ceq F(\phi)$. +\end{Result} + +\begin{Result} +\Item{2.} If $f - \phi$ is positive for $x > x_{0}$, we can determine an increasing function~$F$ +such that $F(f) \cgt F(\phi)$. +\end{Result} + +\begin{Result} +\Item{3.} If $f - \phi$ is monotonic and not negative for $x > x_{0}$, and $F(f) \ceq F(\phi)$, +however great the increase of~$F$, then $f = \phi$ from a certain value of~$x$ onwards. +\end{Result} + +{\Loosen To these he may add the theorem (analogous to that proved at the end of~§\;7) +that \begin{Result}$f \cgt \phi$ involves $F(f) \cgt F(\phi)$ if $\log F(x)/\log x$ is an increasing +function\end{Result} (a condition which may for practical purposes be replaced by +$F \cgt x$).} + +\Paragraph{9.} Let us consider some examples of the theorems of the last paragraph. + +\Item{(i)} Let $f = x^{m}$ ($m > 1$) and $\phi = x$. Then, following the argument of §\;8~(1), +we have $\theta(\phi) = \phi^{m}$. We may take +\[ +a = e, \qquad +a_{1} = e^{m}, \qquad +a_{2} = e^{m^{2}},\ \dots, \qquad +a_{n} =e^{m^{n}},\ \dots, +\] +and we have to define~$F$ so that +\[ +F(e^{m^{n}}) = \tfrac{1}{2}nK. +\] +The most natural solution of this equation is +\[ +F(x) = K\log\log x/2\log m. +\] +And in fact +\[ +F(x^{m}) - F(x) = \frac{K}{2\log m}\{\log(m\log x) - \log\log x\} + = \tfrac{1}{2}K, +\] +so that $x^{m} \ceq x\ (F)$. +%% -----File: 024.png---Folio 16------- + +\Item{(ii)} Let $f = e^{x} + e^{-x}$, $\phi = e^{x}$. Following the argument of §\;8~(2), we find +\[ +\lambda = e^{-x} = \bar{\lambda}, \qquad +\mu(\phi) = 1/\phi, +\] +and we may take $\varpi(\phi) = \phi^{1+\alpha}$ ($\alpha > 0$). This makes $F$ a constant multiple of~$x^{2+\alpha}$, +and it is easy to verify that +\[ +(e^{x} + e^{-x})^{k} - e^{kx} \to \infty, +\] +if $k > 2$. + +\Item{(iii)} The relation $F(f) \ceq F(\phi)$ is equivalent to $f \ceq \phi\ (\log F)$. Using +the result of~(i) we see that $F(x^{m}) \ceq F(x)$ if $F \cleq \log x$. Similarly, using the +result of~(ii), we see that $F(e^{x} + e^{-x}) \cgt F(e^{x})$ if $F \cgeq e^{x^{k}}$ ($k > 2$). +\end{Remark} + +\Paragraph{10.} Before leaving this part of our subject, let us observe that all +of the substance of §§\;1--6 of this section may be extended to the case +in which our symbols $\cgt$,~etc., are defined by reference to an arbitrary +increasing function~$F$. We leave it as an exercise to the reader to +effect these extensions. + +\Chapter{III.}{Logarithmico-Exponential Scales.} + +\Paragraph{1.} \First{The} only scales of infinity that are of any practical importance +in analysis are those which may be constructed by means of the +logarithmic and exponential functions. + +We have already seen (\Ref{ii.}{§\;3}) that +\[ +e^{x} \cgt x^{n} +\] +for any value of~$n$ however great. From this it follows that +\[ +\log x \clt x^{1/n} +\] +for any value of $n$.\footnote + {It was pointed out above (\Ref{ii.}{§\;5}) that $\phi \cgt \bar{\phi}$ does not necessarily involve $\psi \clt \bar{\psi}$ + ($\psi$,~$\bar{\psi}$ being the functions inverse to $\phi$,~$\bar{\phi}$). But it does involve $\psi < \bar{\psi}$ for sufficiently + large values of~$x$, and therefore $\psi \cleq \bar{\psi}$. Hence $\phi \cgt \phi_{n}$ (for any~$n$) involves $\psi \cleq \psi_{n}$ + (for any~$n$) and therefore, if $(\psi_{n})$~is a descending scale, as is in this case obvious, + $\psi \clt \psi_{n}$ for any~$n$. For proofs of the relations $e^{x} \cgt x^{n}$, $\log x \clt x^{1/n}$, proceeding on + different lines, see my \textit{Course of pure mathematics}, pp.~345,~350.} + +It is easy to deduce that +\begin{gather*} +e^{e^{x}} \cgt e^{x^{n}}, \qquad +e^{e^{e^{x}}} \cgt e^{e^{x^{n}}},\ \dots, \\ +\log\log x \clt (\log x)^{1/n}, \qquad +\log\log\log x \clt (\log\log x)^{1/n},\ \dots. +\end{gather*} +%% -----File: 025.png---Folio 17------- + +The repeated logarithmic and exponential functions are so important +in this subject that it is worth while to adopt a notation for +them of a less cumbrous character. We shall write +\begin{alignat*}{3} +%[** TN: Unaligned in the original] +l_{1}x &\eqq lx \eqq \log x, \qquad& +l_{2}x &\eqq llx, \qquad& +l_{3}x &\eqq ll_{2}x,\ \dots,\\ +e_{1}x &\eqq ex \eqq e^{x}, \qquad& +e_{2}x &\eqq eex, \qquad& +e_{3}x &\eqq ee_{2}x,\ \dots. +\end{alignat*} + +It is easy, with the aid of these functions, to write down any +number of ascending scales, each containing only functions whose +increase is greater than that of any function in any preceding scale; +for example +\begin{gather*} +x,\quad x^{2},\ \dots,\quad x^{n},\ \dots;\qquad +e^{x},\quad e^{2x},\ \dots,\quad e^{nx},\ \dots; \\ +e^{x^{2}},\quad e^{x^{3}},\ \dots,\quad e^{x^{n}},\ \dots;\qquad +e_{2}x,\quad e_{3}x,\ \dots,\quad e_{n}x,\ \dots. +\end{gather*} + +In among the functions of these scales we can of course interpolate +new functions as freely as we like, using, for instance, such functions as +\[ +x^{\alpha} e^{\beta x^{\gamma} e^{\delta x^{\epsilon}}}, +\] +where $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$ are any positive numbers; and we can of course +construct non-enumerable (\Ref{ii.}{§\;1}) as well as enumerable scales. +Similarly we can construct any number of descending scales, each +composed of functions whose increase is less than that of any functions +in any preceding scale: for example +\[ +lx, \quad (lx)^{1/2}, \ \dots, \quad (lx)^{1/n},\ \dots; \qquad +l_{2}x, \quad l_{3}x, \ \dots, \quad l_{n}x,\ \dots. +\] + +Two special scales are of particularly fundamental importance; the +ascending scale +\[ +\LTag{(E)} +x, \quad ex, \quad e_{2}x, \quad e_{3}x, \ \dots, +\] +and the descending scale +\[ +\LTag{(L)} +x, \quad lx, \quad l_{2}x, \quad l_{3}x, \ \dots. +\] + +These scales mark the \emph{limits} of all logarithmic and exponential +scales: it is of course, in virtue of the general theorems of~\Ref{ii.}{}, possible +to define functions whose increase is more rapid than that of any~$e_{n}x$ +or slower than that of any~$l_{n}x$; but, as we shall see in a moment, +this is \emph{not} possible if we confine ourselves to functions defined by +a finite and explicit formula involving only the ordinary functional +symbols of elementary analysis. + +\Paragraph{2.} We define a \emph{logarithmico-exponential function} (shortly, an +\emph{$L$-function}) as a real one-valued function defined, for all values of~$x$ +greater than some definite value, by a finite combination of the +ordinary algebraical symbols (viz.\ $+$,~$-$, $×$,~$÷$,~$\sqrt[n]{}$) and the functional +symbols $\log(\dots)$ and $e^{(\dots)}$, operating on the variable~$x$ and on real +constants. +%% -----File: 026.png---Folio 18------- + +\begin{Remark} +It is to be observed that the result of working out the value of the +function, by substituting~$x$ in the formula defining it, is to be real at all +stages of the work. It is important to exclude such a function +\[ +\tfrac{1}{2}\{e^{\sqrt{-x^{2}}} + e^{-\sqrt{-x^{2}}}\}, +\] +which, with a suitable interpretation of the roots, is equal to~$\cos x$. +\end{Remark} + +\begin{Theorem} +Any $L$-function is ultimately continuous, of constant +sign, and monotonic, and, as $x \to \infty$, tends to~$\infty$, or to zero or to some +other definite limit. Further, if $f$~and~$\phi$ are $L$-functions, one or other +of the relations +\[ +f \cgt \phi, \qquad +f \ceqq \phi, \qquad +f \clt \phi +\] +holds between them. +\end{Theorem} + +We may classify $L$-functions as follows, by a method due to +Liouville.\footnote + {See my tract \textit{The integration of functions of a single variable} (No.~2 of this + series), pp.~5 \textit{et~seq.}, where references to Liouville's original memoirs are given.} +An $L$-function is of order zero if it is purely algebraical; +of order~$1$ if the functional symbols $l(\dots)$ and $e(\dots)$ which occur +in it bear only on algebraical functions; of order~$2$ if they bear only +on algebraical functions or $L$-functions of order~$1$; and so on. Thus +\[ +x^{x^{x}} = e^{\log x e^{x\log x}} +\] +is of order~$3$. As the results stated in the theorem are true of +algebraical functions, it is sufficient to prove that, if true of $L$-functions +of order $n - 1$, they are true of $L$-functions of order~$n$. + +Let us observe first that if $f$~and~$\phi$ are $L$-functions, so is~$f/\phi$. +Hence the last part of the theorem is a mere corollary of the first +part. Again, the derivative of an $L$-function of order~$n$ is an $L$-function +of order~$n$ (or less). Hence it is enough to prove that, if +the results stated are true of $L$-functions of order~$n - 1$, then an +$L$-function of order~$n$ is ultimately continuous and of constant sign, +\ie\ that it is continuous and cannot vanish for a series of values of~$x$ +increasing beyond limit. For, if this is true of any $L$-function of +order~$n$, it is true of the derivative of any such function; and therefore +the function itself is ultimately continuous and monotonic. + +Now any $L$-function of order~$n$ can be expressed in the form +\begin{align*} +f_{n} &= A\{e\phi_{n-1}^{(1)}, e\phi_{n-1}^{(2)}, \dots, e\phi_{n-1}^{(r)},\ + l\psi_{n-1}^{(1)}, \dots, l\psi_{n-1}^{(s)}, + \chi_{n-1}^{(1)}, \dots, \chi_{n-1}^{(t)}\}\\ + &= A\{z_{1}, z_{2}, \dots, z_{q}\}, +\end{align*} +say, where $q = r + s + t$, the functions with suffix~$n - 1$ are $L$-functions +of order~$n - 1$, and $A$~denotes an algebraical function: and there is +therefore an identical relation +\[ +F \eqq M_{0} f_{n}^{p} + M_{1} f_{n}^{p-1} + \dots + M_{p} = 0, +\] +%% -----File: 027.png---Folio 19------- +where the coefficients are polynomials in $z_{1}$,~$z_{2}$, \dots,~$z_{q}$. These polynomials +are comprised in the class of functions +\[ +M = \sum \rho_{n-1} e\sigma_{n-1} (l\tau_{n-1}^{(1)})^{\kappa_{1}} (l\tau_{n-1}^{(2)})^{\kappa_{2}} \dots (l\tau_{n-1}^{(h)})^{\kappa_{h}}, +\] +in which the $\kappa$'s are positive integers, the number of terms in the +summation is finite, and the functions with suffix~$n - 1$ are again +$L$-functions of order~$n - 1$. So also are +\[ +\frac{dM_{0}}{dx}, \quad +\frac{dM_{1}}{dx},\ \dots, \quad +\frac{dM_{p}}{dx}, +\] +and the discriminant of~$F$ \textit{qua} function of~$f_{n}$. + +Let us suppose our conclusions established in so far as relates to +functions of the type~$M$. Then it follows by a well known theorem\footnote + {If $F(x, y)$ is a function of $x$~and~$y$ which vanishes for $x = a$, $y = b$, and has + derivatives $\dfrac{\dd F}{\dd x}$,~$\dfrac{\dd F}{\dd y}$ continuous about~$(a, b)$, and if $\dfrac{\dd F}{\dd y}$~does not vanish for $x = a$, + $y = b$, then there is a unique continuous function~$y$ which is equal to~$b$ when $x = a$, + and satisfies the equation $F(x, y) = 0$ identically. See, \eg, W.~H.~Young, \textit{Proc.\ + Lond.\ Math.\ Soc.}, vol.~7, pp.~397 \textit{et~seq.}} +that $f_{n}$~is continuous, and, since $f_{n} = 0$ involves $M_{p} = 0$, that $f_{n}$~also is +ultimately of constant sign. + +Hence it is enough to establish our conclusions for functions of the +type~$M$. Let us call +\[ +\kappa_{1} + \kappa_{2} + \dots + \kappa_{h} +\] +the \emph{degree} of a term of~$M$, and let us suppose that the greatest degree +of a term of~$M$ is~$\lambda$, and that there are $\mu$~terms of degree~$\lambda$, and that +the term printed in the expression of~$M$ above is one of them. + +In the first place it is obvious, from the form of~$M$ and the fact +that $ey$~and~$ly$ are ultimately continuous when $y$~is ultimately continuous +and monotonic, that $M$~is ultimately continuous. Again, if +$M$~vanishes for values of~$x$ surpassing all limit, the same is true of +\[ +M/(\rho_{n-1} e\sigma_{n-1}), +\] +and therefore, by Rolle's theorem,\footnote + {If a function possesses a derivative for all values of its argument, the + derivative must have at least one root between any two roots of the function + itself.} +of the derivative of the latter +function. But the reader will easily verify that when we differentiate, +and arrange the terms of the derivative in the same manner as those +of~$M$, we obtain a function of the same form as~$M$ but containing at +most $\mu - 1$~terms of order~$\lambda$. And by repeating this process we clearly +arrive ultimately at a function of the form +\[ +N = \sum \rho_{n-1} e\sigma_{n-1}, +\] +%% -----File: 028.png---Folio 20------- +in which there are no factors of the form~$l\tau_{n-1}$, and which must vanish +for a sequence of values of~$x$ surpassing all limit. Hence it is +sufficient for our purpose to prove that this is impossible. + +Let the number of terms in~$N$ be~$\varpi$. Then +\[ +\frac{d}{dx} \{N/(\rho_{n-1} e\sigma_{n-1})\} +\] +must (for reasons similar to those advanced above) vanish for values +of~$x$ surpassing all limit. But when we differentiate, and arrange +the terms of the derivative in the same manner as those of~$N$, we +are left with a function of the same form as~$N$, but containing only +$\varpi - 1$~terms. And it is clear that a repetition of this process leads to +the conclusion that a function of the type +\[ +\rho_{n-1} e\sigma_{n-1} +\] +vanishes for values of~$x$ surpassing all limit, which is \textit{ex~hypothesi} +untrue. Hence the theorem is established. + +\Paragraph{3.} The proof just given, it may be observed, does not in any way +depend upon the fact that the symbols of algebraical functionality, +admitted into the definition of $L$-functions, are of an \emph{explicit} character. +We might admit such functions as +\[ +e_{2}\sqrt{ly}, +\] +where $y^{5} + y - x = 0$. But the case contemplated in the definition +seems to be the only one of any interest. + +Another interesting theorem is: \begin{Result}if $f$~is any $L$-function, we can find +an integer~$k$ such that +\[ +f \clt e_{k}x; +\] +and, if $f \cgt 1$, we can find~$k$ so that +\[ +f \cgt l_{k}x: +\] +that is to say, an $L$-function cannot increase more rapidly than any +exponential, or more slowly than any logarithm. +\end{Result} + +More precisely, an $L$-function of order~$n$ cannot satisfy $f \cgt e_{n}(x^{\Delta})$ +or $1 \clt f \clt (l_{n}x)^{\delta}$. The first part of this result is easily established; +the second appears to require a more elaborate proof. + +\Paragraph{4.} Let $f$~and~$\phi$ be any two $L$-functions which tend to infinity +with~$x$, and let $\alpha$ be any positive number. Then one of the three +relations +\[ +f \cgt \phi^{\alpha}, \qquad +f \ceqq \phi^{\alpha}, \qquad +f \clt \phi^{\alpha} +\] +must hold between $f$ and~$\phi$; and the second can hold for at most one +%% -----File: 029.png---Folio 21------- +value of~$\alpha$. If the first holds for any~$\alpha$ it holds for any smaller~$\alpha$; and +if the last holds for any~$\alpha$ it holds for any greater~$\alpha$. + +Then there are three possibilities. Either the first relation holds +for every~$\alpha$; then +\[ +f \cgt \phi^{\Delta}. +\] +Or the third holds for every~$\alpha$; then +\[ +f \clt \phi^{\delta}. +\] +Or the first holds for some values of~$\alpha$ and the third for others; and +then there is a value a of~$\alpha$ which divides the two classes of values of~$\alpha$, +and we may write +\[ +f = \phi^{\alpha} f_{1}, +\] +where $\phi^{-\delta} \clt f_{1} \clt \phi^{\delta}$. We shall find this result very useful in the +sequel. + +\Chapter[Logarithmico-Exponential Scales.] +{IV.}{Special Problems Connected with Logarithmico-Exponential Scales.} + +\begin{Remark} +\Paragraph{1. The functions $e_{r}(l_{s}x)^{\mu}$.} We have agreed to express the fact that, +however large be~$\alpha$ and however small be~$\beta$, $x^{\alpha}$~has an increase less than that +of~$e^{x^{\beta}}$, by +\[ +\Tag{(1)} +x^{\Delta} \clt e^{x^{\delta}}.\footnote + {Such a relation as + \[ + x^{\Delta_{1}} (lx)^{\Delta_{2}} \clt e^{\delta_{1} x^{\delta_{2}} (lx)^{-\Delta_{3}}} + \] + might at first sight appear to afford more information than~\Eq{(1)}: but + \[ + x^{\Delta_{1}} (lx)^{\Delta_{2}} \clt x^{\Delta_{1}'}, \qquad + \delta_{1} x^{\delta_{2}} (lx)^{-\Delta_{3}} \cgt x^{\delta_{2}'}, + \] + where $\Delta_{1}'$,~$\delta_{2}'$ are any positive numbers greater than~$\Delta_{1}$ and less than~$\delta_{2}$ respectively. + Hence our relation really expresses no more than~\Eq{(1)}.} +\] + +Let us endeavour to find a function~$f$ such that +\[ +x^{\Delta} \clt f \clt e^{x^{\delta}}. +\Tag{(2)} +\] + +If $\phi_{1} \cgt \phi_{2}$, $e^{\phi_{1}} \cgt e^{\phi_{2}}$ (\Ref{ii.}{§\;8}). Thus \Eq{(2)}~will certainly be satisfied if +\[ +\log x \clt \log f \clt x^{\delta}. +\] +Hence a solution of our problem is given by +\[ +f = e^{(\log x)^{1+\delta}}. +\] +%% -----File: 030.png---Folio 22------- + +Similarly we can prove that +\[ +f = e^{(\log x)^{1-\delta}} +\] +satisfies +\[ +(\log x)^{\Delta} \clt f \clt x^{\delta}. +\] + +It will be convenient to write +\[ +e_{0}x \eqq l_{0}x \eqq x, +\] +and then we have the relations +\[ +e_{0}(l_{1}x)^{\gamma} + \clt e_{1}(l_{1}x)^{1-\delta} + \clt e_{0}(l_{0}x)^{\gamma} + \clt e_{1}(l_{1}x)^{1+\delta} + \clt e_{1}(l_{0}x)^{\gamma}, +\Tag{(3)} +\] +where $\gamma$~denotes \emph{any} positive number.\footnote + {Here $\delta$, as usual, denotes `any positive number however small.' Of course, in + using the index~$1 - \delta$, it is tacitly implied that $\delta < 1$.} + +Let us now consider the functions +\[ +f = e_{r}(l_{s}x)^{\mu}, \qquad +f' = e_{r'}(l_{s'}x)^{\DPtypo{\mu}{\mu'}}, +\] +where $\mu$,~$\mu'$ are positive and not equal to~$1$. If $r = r'$, $f \cgt f'$ or $f \clt f'$ according +as $s < s'$ or $s > s'$. If $s = s'$, the same relations hold according as $r > r'$ or $r < r'$. +If $r = r'$ and $s = s'$, then $f \cgt f'$ or $f \clt f'$ according as $\mu > \mu'$ or $\mu < \mu'$. Leaving +these cases aside, suppose $s > s'$, $s - s' = \sigma > 0$. Putting $l_{s'}x = y$, we obtain +\[ +f = e_{r}(l_{\sigma}y)^{\mu}, \qquad +f' = e_{r'}y^{\mu'}. +\] +If $r < r'$ it is clear that $f \clt \phi$. If $r > r'$, let $r - r' = \rho$; then +\[ +l_{r}f = (l_{\sigma}y)^{\mu}, \qquad +l_{r}f' = l_{\rho}y^{\mu'} \ceqq l_{\rho}y: +\] +if $\rho > 1$ the symbol~$\ceqq$ may be replaced by~$\sim$. If $\sigma > \rho$, $l_{r}f \clt l_{r}f'$ and so +$f \clt f'$. If $\sigma < \rho$, $f \cgt f'$. If $\sigma = \rho$, $f \cgt f'$ or $f \clt f'$ according as $\mu > 1$ or +$\mu < 1$. Thus +\[ +f \cgt f' \quad (r - s > r' - s'), \qquad +f \clt f' \quad (r - s < r' - s'), +\] +while if $r - s = r' - s'$, $f \cgt f'$ or $f \clt f'$ according as $\mu > 1$ or $\mu < 1$, $\mu$~being the +exponent of the logarithm of higher order which occurs in $f$~or~$f'$. + +From this it follows that +\begin{gather*} +\dots e_{1}(l_{2}x)^{1-\delta} \clt e_{0}(l_{1}x)^{\gamma} \eqq (lx)^{\gamma} \clt e_{1}(l_{2}x)^{1+\delta} \clt e_{2}(l_{3}x)^{1+\delta} \clt \dots\\ +\dots \clt e_{2}(l_{2}x)^{1-\delta} \clt e_{1}(l_{1}x)^{1-\delta} \clt e_{0}(l_{0}x)^{\gamma} \eqq x^{\gamma} \clt e_{1}(l_{1}x)^{1+\delta} \clt \dots\\ +\dots \clt e_{3}(l_{2}x)^{1-\delta} \clt e_{2}(l_{1}x)^{1-\delta} \clt e_{1}(l_{0}x)^{\gamma} \eqq ex^{\gamma} \clt e_{2}(l_{1}x)^{1+\delta} \clt \dots\DPtypo{}{.} +\end{gather*} + +These relations enable us to interpolate to any extent among what we may +call the fundamental logarithmico-exponential orders of infinity, viz.\ $(l_{k}x)^{\gamma}$, +$x^{\gamma}$, $e_{k}x^{\gamma}$. Thus +\[ +e^{(lx)^{1+\delta}}, \quad +e^{e^{(llx)^{1+\delta}}},\ \dots, +\] +and +\[ +e^{e^{(lx)^{1-\delta}}}, \quad +e^{e^{e^{(llx)^{1-\delta}}}},\ \dots, +\] +are two scales, the first rising from above~$x^{\gamma}$, the second falling from below~$ex^{\gamma}$, +and never overlapping. + +These scales, and the analogous scales which can be interpolated between +other pairs of the fundamental logarithmico-exponential orders, possess +%% -----File: 031.png---Folio 23------- +another interesting property. The two scales written above \begin{Result}cover up \emph{(to put +it roughly)} the whole interval between $x^{\gamma}$ and~$ex^{\gamma}$, so far as $L$-functions \(\Ref{iii.}{§\;2}\) +are concerned\end{Result}: that is to say, it is impossible that an $L$-function~$f$ should +satisfy +\begin{alignat*}{2} +f &\cgt e_{r}(l_{r}x)^{1+\delta}, &&\RTag{(\emph{every} $r$),}\\ +f &\clt e_{r+1}(l_{r}x)^{1-\delta},&&\RTag{(\emph{every} $r$);} +\end{alignat*} +and the corresponding pairs of scales lying between $(l_{k+1}x)^{\gamma}$ and~$(l_{k}x)^{\gamma}$, or +between $e_{k}x^{\gamma}$ and~$e_{k+1}x^{\gamma}$, possess a similar property. This property is +analogous to that possessed (\Ref{iii.}{§\;3}) by the scales $(l_{r}x)$,~$(e_{r}x)$; viz.\ that no +$L$-function~$f$ can satisfy $f \cgt e_{r}x$, or $1 \clt f \clt l_{r}x$, for all values of~$r$. A little +consideration is all that is needed to render this theorem plausible: to +attempt to carry out the details of a formal proof would occupy more space +than we can afford. + +\Paragraph{2.} \begin{Example}\Item{(i)} Compare the rates of increase of +\[ +f = (lx)^{(lx)^{\mu}}, \qquad +\phi = x^{(lx)^{-\nu}}. +\] +\end{Example} + +These functions are the same as $e\{(lx)^{\mu}\, llx\}$,~$e\{(lx)^{1-\nu}\}$. If $\mu + \nu \geqq 1$, $f \cgt \phi$; +if $\mu + \nu < 1$, $f \clt \phi$. + +\begin{Example}\Item{(ii)} Compare the rates of increase of +\[ +f = x^{a}(lx)^{b}, \qquad +\phi = e^{A(lx)^{\alpha}(llx)^{\beta}}, \qquad +(a,\ A,\ \alpha > 0). +\] +\end{Example} +Here $f = e(a\, lx + b\, llx)$. If $\alpha < 1$, then $f \cgt \phi$; if $\alpha > 1$, then $f \clt \phi$. If $\alpha = 1$, +$\beta < 0$, then $f \cgt \phi$; if $\alpha = 1$, $\beta > 0$, then $f \clt \phi$. If $\alpha = 1$, $\beta = 0$, $a > A$, then +$f \cgt \phi$; if $\alpha = 1$, $\beta = 0$, $a < A$, then $f \clt \phi$. If $\alpha = 1$, $\beta = 0$, $a = A$, then $f \cgt \phi$ +if $b > 0$ and $f \clt \phi$ if $b < 0$. Finally if $\alpha = 1$, $\beta = 0$, $a = A$, $b = 0$ the two functions +are identical. + +\begin{Example}\Item{(iii)} Compare the increase of $f = x^{\phi/(1+\phi)}$, where $\phi$~is a function of~$x$ such +that $\phi \cgt 1$, with that of~$x^{\gamma}$. +\end{Example} + +It is clear that $f \cleq x$, but $f \cgt x^{\gamma}$ for any value of~$\gamma$ less than unity. For, +if $x$~is large enough, $\phi > n$, where $n$~is any positive integer, and so +\[ +f > x^{n/(1+n)}. +\] +Again $f = xe^{-lx/(1+\phi)}$, and so, if $\phi \clt lx$, $f \clt x$: but if $\phi \ceq lx$, $f \ceq x$; while if +$\phi \cgt lx$, $f \sim x$. + +\Paragraph{3. Successive approximations to a logarithmico-exponential function.} +Consider such a function as +\[ +f = \sqrt{x}(lx)^{2} e^{\sqrt{lx}(l_{2}x)^{2}e^{\sqrt{l_{2}x}(l_{3}x)^{2}}}. +\] +If we omit one or more of the parts of the expression of~$f$ we obtain another +function whose increase differs more or less widely from that of~$f$. The +question arises as to which parts are of the greatest and which of the least +importance; \ie\ as to which are the parts whose omission affects the increase +of~$f$ most or least fundamentally. + +Taking logarithms we find +\[ +\lf = \tfrac{1}{2}lx + \sqrt{lx}(l_{2}x)^{2} e^{\sqrt{l_{2}x}(l_{3}x)^{2}} + 2l_{2}x, +\Tag{(a)} +\] +%% -----File: 032.png---Folio 24------- +the three terms being arranged in order of importance. Again +\[ +l_{2}f = l_{2}x - l2 + \epsilon, \qquad +l_{3}f = l_{3}x + \epsilon, +\] +where (\Ref{i.}{§\;5}) in each of the last equations $\epsilon$~denotes a function (not the +same function) which tends to zero as $x \to \infty$. If we neglect this term in +each of them in turn we deduce the approximations +\[ +\ITag{(1)} f = x,\qquad +\ITag{(2)} f = \sqrt{x}. +\] + +By neglecting the last term in the equation~\Eq{(a)} we obtain the much closer +approximation +\[ +\ITag{(6)} f = \sqrt{x} e^{\sqrt{lx} (l_{2}x)^{2} e^{\sqrt{l_{2}x} (l_{3}x)^{2}}}. +\] + +In order to obtain a more complete series of approximations to~$f$ we must +replace the equation~\Eq{(a)} by a series of approximate equations. Now if +\[ +\phi = \sqrt{lx}(l_{2}x)^{2} e^{\sqrt{l_{2}x} (l_{3}x)^{2}} +\] +we have +\begin{gather*} +l\phi = \tfrac{1}{2}l_{2}x + \sqrt{l_{2}x} (l_{3}x)^{2} + 2l_{3}x,\\ +l_{2}\phi = l_{3}x - l2 + \epsilon, \qquad +l_{3}\phi = l_{4}x + \epsilon. +\end{gather*} +Hence we obtain (0)~$\phi = lx$, (3)~$\phi = \sqrt{lx}$, and (5)~$\phi = \sqrt{lx} e^{\sqrt{l_{2}x}(l_{3}x)^{2}}$ as +approximations to the increase of~$\phi$: of these, however, the first is valueless, +inasmuch as it would make~$\phi$ preponderate over the first term on the right +hand side of~\Eq{(a)}. + +A similar argument, applied to the function $e^{\sqrt{l_{2}x}(l_{3}x)^{2}}$, leads us to interpolate +(4)~$\phi = \sqrt{lx} e^{\sqrt{l_{2}x}}$ between (3)~and~(5). We can now, by adopting +a series of approximate forms of the equation~\Eq{(a)}, deduce a complete system +of closer and closer approximations to the increase of~$f$, viz.\ +\begin{gather*} +\ITag{(1)} x,\qquad +\ITag{(2)} \sqrt{x},\qquad +\ITag{(3)} \sqrt{x} e^{\sqrt{lx}},\qquad +\ITag{(4)} \sqrt{x} e^{\sqrt{lx} e^{\sqrt{l_{2}x}}},\\ +\ITag{(5)} \sqrt{x} e^{\sqrt{lx} e^{\sqrt{l_{2}x}(l_{3}x)^{2}}},\qquad +\ITag{(6)} \sqrt{x} e^{\sqrt{lx}(l_{2}x)^{2} e^{\sqrt{l_{2}x}(l_{3}x)^{2}}}. +\end{gather*} +This order corresponds exactly to the order of importance of the various parts +of the expression of~$f$. + +\Paragraph{4. Legitimate and illegitimate forms of approximation to a logarithmico-exponential +function.} In applications of this theory, such as +occur, for instance, in the theory of integral functions, we are continually +meeting such equations as +\[ +f = (1 + \epsilon)e^{x^{\alpha}}, \qquad +f = e^{(1+\epsilon)x^{\alpha}}, \qquad +f = e^{x^{\alpha+\epsilon}}, \qquad (\alpha > 0). +\Tag{(1)} +\] + +It is important to have clear ideas as to the degree of accuracy of such +representations of~$f$. The simplest method is to take logarithms repeatedly, +as in §\;3~above. + +In the first example the term~$\epsilon$ does not affect the increase of~$f$: we have +$f \sim ex^{\alpha}$. This is not true in the second; but $\lf \sim x^{\alpha}$, so that the term~$\epsilon$ does +not affect the increase of~$\lf$; while in the third this is not true, though $\llf \sim \alpha$. +Of the three formulae the first gives the most, and the last the least, information +as to the increase of~$f$ (see also \Ref{vii.}{§\;3}). +%% -----File: 033.png---Folio 25------- + +Such a formula as +\[ +f = xe^{(1+\epsilon)x^{\alpha}} +\Tag{(2)} +\] +would not be a legitimate form of approximation at all. For the factor~$e(\epsilon x^{\alpha})$ +which is not completely specified may well be far more important than the +explicitly expressed factor~$x$: we might for example have $\epsilon = x^{-\beta}$, where +$0 < \beta < \alpha$, in which case $e(\epsilon x^{\alpha})$ is more important than any power of~$x$. Thus +\Eq{(2)}~does not really convey more information than the second equation~\Eq{(1)}, +and to use it would involve a logical error similar to that involved in saying +that the sun's distance is $92,713,600$~miles, with a probable error of some +$100,000$~miles. + +\Paragraph{5. Attempts to represent orders of infinity by symbols.} It is +natural to try to devise some simple method of representing orders of +infinity by symbols which can be manipulated according to laws resembling +as far as possible those of ordinary algebra. Thus Thomae\footnote + {\textit{Elementare Theorie der analytischen Funktionen}, S.~112.} +has proposed +to represent the order of infinity of $f = x^{\alpha}(lx)^{\alpha_{1}}(l_{2}x)^{\alpha_{2}} \dots$ by +\[ +Of = \alpha + \alpha_{1}l_{1} + \alpha_{2}l_{2} + \dots,\footnote + {The reader will not confuse this use of the symbol~$O$ (which does not extend + beyond this paragraph) with that explained in \Ref{i.}{§\;5}.} +\] +where the symbols $l_{1}$,~$l_{2}$,~\dots\ are to be regarded as new units. It is clear that +these units cannot, in relation to one another, obey the Axiom of Archimedes:\footnote + {`If $x > y > 0$, we can find an integer~$n$ such that $ny > x$.'} +however great~$n$, $nl_{2}$~cannot be as great as~$l_{1}$, nor $nl_{1}$~as great as~$1$. + +The consideration of a few simple cases is enough to show that any such +notation, if it is to be of any use, must obey the following laws: +\begin{alignat*}{2} + &\Item{(i)} && \text{if} f \cgeq \phi, \qquad O(f + \phi) = Of;\\ + &\Item{(ii)} &&\quad O(f\phi) = Of + O\phi;\\ + &\Item{(iii)}\quad&&\quad O\{f(\phi)\} = Of × O\phi. +\end{alignat*} + +And Pincherle\footnote + {\lc\ (see \PageRef{p.}{13} above).} +has pointed out that these laws are in any case inconsistent +with the maintenance of the laws of algebra in their entirety. +Thus if +\[ +Ox = 1, \qquad +O\, lx = \lambda, +\] +we have, by~(iii), $O\, llx = \lambda^{2}$, and by (iii)~and~(ii), +\[ +O\, l(x\, lx) = \lambda(1 + \lambda) = \lambda + \lambda^{2}; +\] +and on the other hand, by~(i), +\[ +O\, l(x\, lx) = O(lx + llx) = \lambda. +\] + +Pincherle has suggested another system of notation; but the best yet +formulated is Borel's.\footnote + {\textit{Leçons sur les séries à termes positifs}, pp.~35 \textit{et~seq.}; for further information + see his recently published \textit{Leçons sur la théorie de la croissance}, pp.~14 \textit{et~seq.}} +Borel preserves the three laws (i),~(ii),~(iii), the +%% -----File: 034.png---Folio 26------- +commutative law of addition, and the associative law of multiplication. But +multiplication is no longer commutative, and only distributive on one side.\footnote + {$(a + b)c = ac + bc$, but in general $a(b + c) \neq ab + ac$.} +He would denote the orders of +\begin{align*} +e^{x}x^{n},&& +x^{n}(lx)^{p},&& +e^{2x},&& +e^{x^{2}},&& +e^{e^{x}},&& +e^{\sqrt{lx}},&& +\tfrac{1}{2} x, \\ +\intertext{by} +\omega + n,&& +n + \frac{p}{\omega},&& +2 · \omega,&& +\omega · 2,&& +\omega^{2},&& +\omega · \frac{1}{2} · \frac{1}{\omega},&& +\frac{1}{\omega} · \frac{1}{2} · \omega. +\end{align*} +But little application, however, has yet been found for any such system of +notation; and the whole matter appears to be rather of the nature of +a mathematical curiosity. +\end{Remark} + +\Chapter[Logarithmico-Exponential Scales.] +{V.}{Functions Which do not Conform to any Logarithmico-Exponential Scale.} + +\Paragraph{1.} \First{We} saw in \Ref{i.}{(§\;2)} that, given two increasing functions $\phi$~and~$\psi$ +($\phi \cgt \psi$), we can always construct an increasing function~$f$ which is, for +an infinity of values of~$x$ increasing beyond all limit, of the order of~$\phi$, +and for another infinity of values of~$x$ of the order of~$\psi$. The actual +construction of such functions by means of explicit formulae we left till +later. We shall now consider the matter more in detail, with special +reference to the case in which $\phi$~and~$\psi$ are $L$-functions. + +We shall say that $f$~is an \emph{irregularly increasing} function (\textit{fonction +à croissance irrégulière}) if we can find two $L$-functions $\phi$~and~$\psi$ ($\phi \cgt \psi$) +such that +\[ +f \geq \phi \quad (x = x_{1},\ x_{2},\ \dots), \qquad +f \leq \psi \quad (x = x_{1}',\ x_{2}',\ \dots), +\] +$x_{1}$,~$x_{2}$,~\dots\ and $x_{1}'$,~$x_{2}'$,~\dots\ being any two indefinitely increasing sequences +of values of~$x$. We shall also say that `the increase of~$f$ is irregular' +and that `the logarithmico-exponential scales are \emph{inapplicable} to~$f$.' + +\begin{Remark} +The phrase `\textit{fonction à croissance irrégulière}' has been defined by various +writers in various senses. Borel\footnote + {\textit{Leçons sur les fonctions entières}, p.~107.} +originally defined $f$ to be \textit{à croissance régulière} if +\[ +e^{x^{\alpha-\delta}} < f < e^{x^{\alpha+\delta}}, \RTag{$(x > x_{0})$,} +\] +or in other words if $\llf \sim \alpha lx$ or $\llf \ceqq lx$. + +This definition was of course designed to meet the particular needs of the +%% -----File: 035.png---Folio 27------- +theory of integral functions: and has been made more precise by Boutroux +and Lindelöf,\footnote + {Boutroux, \textit{Acta Mathematica}, t.~28, p.~97; Lindelöf, \textit{Acta Societatis + Fennicae}, t.~31, p.~1. See also Blumenthal, \textit{Principes de la théorie des fonctions + entières d'ordre infini}.} +who use inequalities of the form +\[ +e^{x^{\alpha}(lx)^{\alpha_{1}} \dots (l_{k}x)^{\alpha_{k}-\delta}} < f < +e^{x^{\alpha}(lx)^{\alpha_{1}} \dots (l_{k}x)^{\alpha_{k}+\delta}}. +\] + +All functions which are not \textit{à croissance régulière} for these writers are +included in our class of irregularly increasing functions. +\end{Remark} + +\Paragraph{2.} The logarithmico-exponential scales may fail to give a complete +account of the increase of a function in two different ways. The +function may be of irregular increase, as explained above, and the +scales \emph{inapplicable}: on the other hand they may be, not inapplicable, +but \emph{insufficient} (\textit{en~défaut}). That is to say, although the increase of +the function does not oscillate from that of one $L$-function to that of +another, there may be no $L$-function capable of measuring it. That +such functions exist follows at once from the general theorems of~\Ref{ii}{}. +Thus we can define a function which tends to infinity more rapidly +than any~$e_{r}x$, or more slowly than any~$l_{r}x$: and the increase of such a +function is more rapid or slower than that of any $L$-function (\Ref{iii.}{§\;2}). +Or again, we can (\Ref{ii.}{§\;6}) define a function whose increase is greater +than that of~$e_{r}(l_{r}x)^{1+\delta}$ (any~$r$) and less than that of~$e_{r+1}(l_{r}x)^{1-\delta}$ (any~$r$); +and the increase of such a function (\Ref{iv.}{§\;1}) cannot be equal to that of +any $L$-function. + +We shall now discuss some actual examples of functions for which +the logarithmico-exponential scales are inapplicable or insufficient. + +\Paragraph{3. Irregularly increasing functions.} Functions whose increase +is irregular may be constructed in a variety of ways. + +\begin{Remark} +\Item{(i)} Pringsheim\footnote + {See \textit{Math.\ Annalen}, Bd.~35, S.~347 \textit{et~seq.}\ and \textit{Münchener Sitzungsberichte}, Bd.~26, + S.~605 \textit{et~seq.}} +has used, in connection with the theory of the convergence +of series, functions of an integral variable~$n$ whose increase is +irregular. A simple example of such a function is +\[ +f(n) = 10^{[(\log_{10} n)^{1/\tau}]^{\tau}}, \RTag{$(\tau > 1)$,} +\] +where $[x]$~denotes the integral part of~$x$. It is easily proved, for instance, +when $\tau = 2$, that the increase of~$f(n)$ varies between that of~$n$ and that of +$n · 10^{1-2\sqrt{\log_{10}n}}$. We shall not do more than mention functions of this type. +They are defined, most naturally, as functions of an integral variable~$n$: if we +extend the definition to the continuous variable, the resulting function is +discontinuous. The definition can of course be modified so as to give a +%% -----File: 036.png---Folio 28------- +continuous function of~$x$ with substantially the same properties; but it is +not easy to effect this by a simple, natural, and explicit formula. + +\Item{(ii)} A more natural type of function is given by +\[ +f = \phi \cos^{2} \theta + \psi \sin^{2} \theta, +\] +where $\phi$,~$\psi$,~$\theta$ are increasing $L$-functions. We have to consider what +conditions $\phi$,~$\psi$,~$\theta$ must satisfy in order that $f$~may increase steadily with~$x$. +That its increase oscillates between that of~$\phi$ and that of~$\psi$ is obvious. + +Differentiating, +\[ +f' = \phi' \cos^{2} \theta + \psi' \sin^{2} \theta + 2(\psi - \phi)\theta' \cos \theta \sin \theta. +\] +Suppose $\phi \cgt \psi$: and let us assume that (as will be proved in the next +chapter) relations between $L$-functions involving the symbols $\cgt$,~etc.\ may be +differentiated and integrated. The condition that $f'$~should always be +positive is $\phi'\psi' \cgt (\phi - \psi)^{2}\theta'^{2}$ or $\phi'\psi' \cgt \phi^{2}\theta'^{2}$. \textit{A~fortiori}, since $\phi' \cgt \psi'$, we +must have $\phi' \cgt \phi\theta'$, or $\log\phi \cgt \theta$. Thus $f$~is certainly monotonic if +\[ +\phi \cgt \psi, \qquad +\log\phi \cgt \theta, \qquad +\psi' \cgt \phi^{2}\theta'^{2}/\phi'. +\] +If, \eg, $\theta = x$, we require $\log\phi \cgt x$, which is satisfied, for example, if +$\phi = x^{\alpha} e^{x^{\rho}}$ ($\rho > 1$). It is convenient to write $a + \rho - 1$ for~$\alpha$. Then, since +$\phi' \sim \rho x^{\alpha+\rho-1} e^{x^{\rho}}$, we must have $\psi' \cgt x^{a} e^{x^{\rho}}$; and so +\[ +\psi \cgt \int^{x} t^{a} e^{t^{\rho}}\, dt + = \frac{1}{\rho} \int^{x} t^{a-\rho+1} \frac{d}{dt}\, (e^{t^{\rho}})\, dt + \sim \frac{1}{\rho} x^{a-\rho+1} e^{x^{\rho}}, +\] +as is easily seen on integrating by parts. Thus we may take $\psi = x^{\beta} e^{x^{\rho}}$, +where $\alpha - 2\rho + 2 < \beta < \alpha$. Changing our notation a little we see that +\[ +f = (x^{\gamma+\delta} \cos^{2} x + x^{\gamma-\delta} \sin^{2} x) e^{x^{\rho}} +\] +is monotonic if $0 < \delta < \rho-1$; and the increase of~$f$ obviously oscillates +between that of~$x^{\gamma+\delta} e^{x^{\rho}}$ and that of~$x^{\gamma-\delta} e^{x^{\rho}}$. Similarly it may be shown +that +\[ +f = (e^{\mu x} \cos^{2} x + e^{\nu x} \sin^{2} x) e^{e^{x}} +\] +is monotonic if $\nu < \mu < \nu + 2$;\footnote + {Cf.\ \textit{Messenger of Mathematics}, vol.~31, p.~1.} +and again the increase of~$f$ is irregular. +\end{Remark} + +\Paragraph{4. Irregularly increasing functions (\continued).} We shall +now consider two more general and more important methods for the +construction of irregularly increasing functions. + +\Item{(iii)} Borel\footnote + {See Borel, \textit{Leçons sur les fonctions entières}, pp.~120~\textit{et~seq.}; \textit{Leçons sur les + séries à termes positifs}, pp.~32~\textit{et~seq}. Borel considers the cases only in which + $\psi = e^{x}$, $\phi = e^{x^{2}}$ or~$e^{e^{x}}$; but his method is obviously of general application. The + proof here given is however more general and much simpler.} +has shown how, by means of power series, we may +define functions which increase steadily with~$x$, while their increase +oscillates to an arbitrary extent. +%% -----File: 037.png---Folio 29------- + +Let +\[ +\phi(x) = \sum a_{n}x^{n}, \qquad +\psi(x) = \sum b_{n}x^{n} +\] +be two integral functions of~$x$ with positive coefficients; suppose also +$\phi \cgt \psi$. The increase of $\phi$~and~$\psi$ may be as large as we like (\Ref{ii.}{§\;4}); +but in each case it must be greater than that of any power of~$x$. + +Then we can define a function +\[ +f(x) = \sum c_{n}x^{n}, +\] +where every~$c_{n}$ is equal either to~$a_{n}$ or to~$b_{n}$, in such a way that, for an +infinity of values~$x_{\nu}$ whose limit is infinity, we have $f \sim \phi$, and for a +similar infinity of values~$x_{\nu}'$ we have $f \sim \psi$.\footnote + {By `$f \sim \phi$ for an infinity of values~$x_{\nu}$' we mean of course that $f/\phi \to 1$ as $x \to \infty$ + through this particular sequence of values.} + +Let $(\eta_{\nu})$ be a sequence of decreasing positive numbers whose limit is +zero. Take a positive number~$x_{0}$ such that $\phi(x_{0}) > 1$, $\psi(x_{0}) > 1$, and a +number~$x_{1}$ greater than~$x_{0}$. When $x_{1}$~is fixed we can choose~$n_{1}$ so that +\[ +\sum_{n_{1}}^{\infty} a_{n}x_{1}^{n} < \tfrac{1}{3} \eta_{1}, \qquad +\sum_{n_{1}}^{\infty} b_{n}x_{1}^{n} < \tfrac{1}{3} \eta_{1}, +\] +and so, if $c_{n}$~is either of $a_{n}$,~$b_{n}$ (however the selection may be made for +different values of~$n$), +\[ +\sum_{n_{1}}^{\infty} c_{n}x_{1}^{n} + < \sum_{n_{1}}^{\infty} (a_{n} + b_{n})x_{1}^{n} + < \tfrac{2}{3} \eta_{1}. +\] + +For $0 \leq n < n_{1}$ we take $c_{n} = a_{n}$. Then +\[ +|f(x_{1})-\phi(x_{1})| + < \sum_{n_{1}}^{\infty} (a_{n} + c_{n})x_{1}^{n} + < \eta_{1}, +\] +and so, since $\phi(x_{1}) > 1$, +\[ +\left|\frac{f(x_{1})}{\phi(x_{1})} - 1\right| < \eta_{1}. +\Tag{(1)} +\] + +Now let $x_{2}$ be a number greater than~$x_{1}$; we can suppose $x_{2}$~chosen +so that +\[ +\biggl(\,\sum_{0}^{n_{1}-1} a_{n}x_{2}^{n}\biggr) \bigg/ \psi(x_{2}) < \tfrac{1}{5} \eta_{2}, \qquad +\biggl(\,\sum_{0}^{n_{1}-1} b_{n}x_{2}^{n}\biggr) \bigg/ \psi(x_{2}) < \tfrac{1}{5} \eta_{2}. +\] +When $x_{2}$~is fixed we can choose~$n_{2}$ ($n_{2} > n_{1}$) so that +\[ +\sum_{n_{2}}^{\infty} a_{n}x_{2}^{n} < \tfrac{1}{5} \eta_{2}, \qquad +\sum_{n_{2}}^{\infty} b_{n}x_{2}^{n} < \tfrac{1}{5} \eta_{2}. +\] + +For $n_{1} \leqq n < n_{2}$ we take $c_{n} = b_{n}$. And, however $c_{n}$~be chosen for +$n \geqq n_{2}$, we have +\[ +\sum_{n_{2}}^{\infty} c_{n}x_{2}^{n} + < \sum_{n_{2}}^{\infty} (a_{n} + b_{n})x_{2}^{n} + < \tfrac{2}{5} \eta_{2}. +\] +%% -----File: 038.png---Folio 30------- +Also +\begin{align*} +%[** TN: Not aligned in the original] +|f(x_{2}) - \psi(x_{2})| + &< \sum_{0}^{n_{1}-1} a_{n}x_{2}^{n} + + \sum_{0}^{n_{1}-1} b_{n}x_{2}^{n} + + \sum_{n_{2}}^{\infty} c_{n}x_{2}^{n} + + \sum_{n_{2}}^{\infty} b_{n}x_{2}^{n} \\ + &< \tfrac{2}{5} \eta_{2} \psi(x_{2}) + + \tfrac{3}{5} \eta_{2} + < \eta_{2}\psi(x_{2}), +\end{align*} +and so +\[ +\left|\frac{f(x_{2})}{\psi(x_{2})} - 1\DPtypo{}{\right|} < \eta_{2}. +\Tag{(2)} +\] + +It is plain that, by a repetition of this process, we can find a +sequence $x_{1}$,~$x_{2}$, $x_{3}$,~\dots\ whose limit is infinity, so that +\[ +% [** TN: Semantic \RTags, but using \RTag entails ad hoc spacing] +\left|\frac{f(x_{3})}{\phi(x_{3})} - 1\right| < \eta_{3} +\quad (3),\qquad +% +\left|\frac{f(x_{4})}{\psi(x_{4})} - 1\right| < \eta_{4} +\quad(4),\qquad +\dots; +\] +and our conclusion is thus established. Incidentally we may remark +that not only $f$~itself, but all its derivatives also, are increasing and +continuous. + +It is clear that, if we were given any number of integral functions +$\phi_{1}$,~$\phi_{2}$, \dots,~$\phi_{k}$, with positive coefficients, we could define~$f$ so that +$f/\phi_{s} \to 1$, as $x \to \infty$ through a suitably chosen sequence of values, for +each of the functions~$\phi_{s}$. + +\begin{Remark} +\Item{(iv)} \textbf{Power series with gaps.} There is another method of constructing +irregularly increasing functions by means of power series which, though less +general theoretically than that explained above, is in some ways more +interesting, inasmuch as the functions to which it leads us are of a far +simpler and more natural type. We shall confine ourselves here to explaining +in general terms the general principle of the method and indicating +a few simple examples.\footnote + {For fuller details see Hardy, \textit{Proc.\ Lond.\ Math.\ Soc.}, vol.~2, pp.~332~\textit{et~seq.}; + \textit{Messenger of Mathematics}, vol.~39, p.~28; Borel, \textit{Rendiconti del Circolo Matematico + di Palermo}, t.~23, p.~320; \textit{Leçons sur la théorie de la croissance}, pp.~111~\textit{et~seq.}; + Blumenthal, \textit{Principes de la théorie des fonctions entières d'ordre infini}, pp.~5~\textit{et~seq.}} + +Let +\[ +\phi(x) = \sum a_{n}x^{n} +\Tag{(1)} +\] +be an integral function with positive coefficients: suppose, to fix our ideas, +that the coefficients decrease steadily as $n$~increases. Suppose also that, for +a particular value of~$x$, +\[ +\varpi(x) = a_{\nu} x^{\nu} +\] +is the greatest term of the series. In general one term will be the greatest, +but for certain particular values of~$x$, say $\xi_{1}$,~$\xi_{2}$,~\dots, two consecutive terms +will be equal.\footnote + {We leave aside the possibility, which obviously applies only to particular + cases, of more than two terms being equal.} +%% -----File: 039.png---Folio 31------- + +As $x$~increases, the index~$\nu$ of~$\varpi(x)$ increases, and tends to~$\infty$ with~$n$: it +thus defines a function~$\nu(x)$ such that +\[ +\nu(x) = i,\quad (\xi_{i} < x < \xi_{i+1}). +\] +At the point of discontinuity~$\xi_{i}$, where $\nu(x)$~jumps from $i - 1$ to~$i$, we may +assign to it the value~$i$. When $\nu$~is thus defined for all values of~$x$, or $\varpi(x)$~defines +a function of~$x$ which tends continuously and steadily to~$\infty$ with~$x$. + +The increase of~$\phi$ is obviously at least as great as that of~$\varpi$; it may be +expected to be greater: but it is, in ordinary cases, not so very much +greater---the increase of~$\varpi$ gives a very fair approximation to that of~$\phi$. +Thus, if $\phi(x) = e^{x}$, $a_{n} = 1/n!$, and $\xi_{i} = i$. And for $i < x < i +1$ we have +\[ +e^{i} < \phi < e^{i+1}, \qquad +(1 - \epsilon_{i}) \frac{e^{i}}{\sqrt{2\pi i}} < \varpi < (1 + \epsilon_{i}) \frac{e^{i+1}}{\sqrt{2\pi i}}.\footnote + {The second pair of inequalities are an immediate consequence of Stirling's + theorem, that $i! \sim i^{i+\frac{1}{2}} e^{-i} \sqrt{2\pi}$.}% +\] +Thus $\phi \cgt \varpi$, but $\log\phi \sim \log\varpi$: the difference between the increases of $\phi$ +and~$\varpi$ is small compared with the increases themselves. + +Now let +\[ +f(x) = \sum a_{\chi(n)} x^{\chi(n)}, +\Tag{(2)} +\] +where $\chi(n) \cgt n$: and let $p(x)$ be the function related to~$f$ as $\varpi(x)$~is to~$\phi$. +The laws of increase of~$\varpi(x)$ and of~$p(x)$ may be expected to be very much +the same, for $p(x)$~is defined by a selection from \emph{some} of the terms from \emph{all} +of which $\varpi(x)$~was selected. The increase of~$f(x)$ clearly cannot be greater, +and may be expected to be less, than that of~$\phi(x)$: but it cannot be less than +that of~$p(x)$. Hence we may expect relations of the type +\[ +p \ceq \varpi \clt f \clt \phi.\footnote + {We \emph{must} have $p \cleq \varpi$, $p \cleq f$, $\varpi \cleq \phi$, $f \cleq \phi$.} +\] +Also it is clear that, the more rapidly we suppose $\chi(n)$~to increase, the lower +in the gap between $\varpi$ and~$\phi$ will $f$~sink, and that, if we suppose $\chi$~to increase +with sufficient rapidity, we may expect to find $\varpi \ceq f$, so that the increase of~$f$ +is completely dominated by that of one (variable) term. + +We then shall have +\[ +f(x) \ceq a_{N(x)}x^{N(x)}, +\] +where $N(x)$~is a function of~$x$ which assumes successively each of a series of +integral values~$N_{i}$, so that +\[ +N(x) = N_{i}, \RTag{$(x_{i} \leqq x < x_{i+1})$.\footnotemark} +\] +\footnotetext{$N_{i}$,~$x_{i}$ are, of course, not the same as $\nu_{i}$,~$\xi_{i}$ above.}% +But, as $x$~increases from $x_{i}$ to~$x_{i+1}$, the order of~$a_{N_{i}}x^{N_{i}}$, considered as a +function of~$x$, may vary considerably, since $N_{i}$, though depending on the +%[** TN: Hardy's notation for a closed interval; inconsistent, not modernizing] +interval $(x_{i}, x_{i+1})$, does not depend on the particular position of~$x$ in that +interval. And so it is clear that we are in this way likely to be led to +functions whose increase is irregular in the sense explained in~§\;1. +%% -----File: 040.png---Folio 32------- + +Suppose, for example, that $a_{n} = n^{-n}$, so that +\[ +\phi(x) = \sum \left(\frac{x}{n}\right)^{n} + \sim \sqrt{\frac{2\pi x}{e}} e^{x/e}.\footnote + {See \Ref{ii.}{§\;3}, and the references given in the footnote to \PageRef{p.}{10}. We might + have taken $\phi(x) = e^{x}$, but our choice of~$\phi(x)$ leads to the simplest examples.} +\] + +Here +\[ +\xi_{i} = i\left(1 + \frac{1}{i}\right)^{i+1} \sim ei, +\] +and it is easily shown that $\varpi(x) \ceq e^{x/e}$. + +Now let $\chi(n) = 2^{n}$, so that +\[ +f(x) = \sum \frac{x^{2^{n}}}{2^{n2^{n}}} = \sum v_{n} +\] +say. Then $v_{i-1} = v_{i}$ if $x = 2^{i+1}$, so that $x_{i} = 2^{i+1}$ and $N_{i} = 2^{i}$ for +\[ +2^{i+1} \leqq x < 2^{i+2}. +\] +For this range of values of~$x$, $v_{i}$~is the greatest term; when $x = 2^{i+2}$, $v_{i} = v_{i+1}$. +Further, it is not difficult to show that $f(x) \ceq p(x) = v_{i}$, the behaviour of~$f(x)$ +being dominated by that of its greatest term.\footnote + {We may say roughly that \emph{in general} $f \sim p$---that is to say, $f/p \to 1$ as~$x \to \infty$ + through any sequence of values not falling inside any of certain intervals surrounding + the values~$\xi_{i}$. At a point~$\xi_{i}$, $f/p$~is nearly equal to~$2$.} + +If we put $x = 2^{i+1+\theta}$, where $0 < \theta < 1$, we find +\[ +f(x) \ceq v_{i} = 2^{(1+\theta)2^{i}} = 2^{\alpha x}, +\] +where $\alpha = (1 + \theta)2^{-1-\theta}$. This is a maximum when $1 + \theta = 1/(\log 2)$, when it +is equal to~$1/(e\log 2) = .53\dots$. Hence the increase of~$f(x)$ oscillates (roughly) +between that of~$2^{.53\dots x}$ and~$2^{\frac{1}{2}x + 1}$.\footnote + {The latter function is multiplied by~$2$, as there are two equal terms when + $\theta = 0$ or~$1$.} + +Similar considerations may be applied to the more general series +\[ +\sum \frac{x^{a^{n}}}{b^{na^{n}}}, +\] +where $a$~is an integer greater than unity. This series is derived from $\sum (x/n^{a})^{n}$, +where $\alpha = (\log b)/(\log a)$, by taking $\chi(n) = a^{n}$. Another example of an irregularly +increasing function defined in a similar manner is +\[ +f(x) = \sum \frac{x^{n^{3}}}{(n^{3})!}, +\] +the increase of which oscillates between the increases of~$e^{x}/\sqrt{x}$ and +\[ +x^{-\frac{1}{2}} e^{x-\frac{9}{8}x^{1/3}}.\footnote + {\textit{Messenger of Mathematics}, vol.~39, p.~28.} +\] +These examples are of course typical of a large class of functions. + +Before we leave this subject let us call attention to a point of considerable +%% -----File: 041.png---Folio 33------- +interest suggested by the foregoing examples. In forming the logarithmico-exponential +scales we started from the scale $x$,~$x^{2}$,~\dots\ and then formed the +function~$\sum \dfrac{x^{n}}{n!}$. If we had started, as we equally well might have done, from +the scale $x^{2}$,~$x^{4}$, $x^{8}$,~\dots\ (cf.~\Ref{ii.}{§\;1}), we should have been led to choose, as a +function transcending this scale, not~$e^{x}$ but some such function as +\[ +\sum \frac{x^{2^{n}}}{(2^{n})!}. +\] +\emph{This is one of the irregularly increasing functions of the type just considered.} +Had we proceeded thus, and completed the construction of our fundamental +scales on similar lines, our fundamental functions would for the most part +have been among those which do not conform to the logarithmico-exponential +scale, and it would have been the functions of that scale that would have +appeared as irregularly increasing functions. +\end{Remark} + +\Paragraph{5. Functions which transcend the logarithmico-exponential +scales.} We turn our attention now to functions for which +the logarithmico-exponential scales are not inapplicable but \emph{insufficient} +(§\;2). Of the existence of such functions we are already assured. +Thus a function which assumes the values $e_{1}(1)$,~$e_{2}(2)$, \dots,~$e_{\nu}(\nu)$,~\dots\ for +$x = 1$, $2$,~\dots, $\nu$,~\dots\ certainly has an increase greater than that of any +logarithmico-exponential function. No such function, however, has as +yet made its appearance naturally in analysis; it will be sufficient, +therefore, to mention two examples of such functions which transcend +the logarithmico-exponential scales in quite different manners. + +\Item{(i)} The series +\[ +\sum \frac{e_{\nu}(x)}{e_{\nu}(\nu)} +\] +has obviously, if it converges, an increase greater than that of any~$e_{\nu}(x)$. +Suppose $k - 1 \leqq x < k$. Then +\[ +\frac{e_{k}(x)}{e_{k}(k)} < 1, \qquad +\frac{e_{k+\nu}(x)}{e_{k+\nu}(k+\nu)} + < \frac{e_{k+\nu}(k)}{e_{k+\nu}(k+\nu)} + < \frac{e_{k+\nu}(k)}{e_{k+\nu}(k+1)}. +\] +But, by the Mean Value Theorem, +\[ +e_{k+\nu}(k+1) = e_{k+\nu}(k) + e_{k+\nu}(y)e_{k+\nu-1}(y) \dots e_{2}(y)e_{1}(y), +\] +where $y$~is some number between $k$~and~$k + 1$; and so +\[ +e_{k+\nu}(k+1) > e_{k+\nu}(k)e_{k+\nu-1}(k) \dots e_{1}(k). +\] +It follows that the terms of the series +\[ +\sum_{\nu=k}^{\infty} \frac{e_{\nu}(x)}{e_{\nu}(\nu)} +\] +are less than those of the series +\[ +1 + \sum_{\nu=1}^{\infty} \frac{1}{e_{1}(k)e_{2}(k) \dots e_{k+\nu-1}(k)}, +\] +%% -----File: 042.png---Folio 34------- +which is plainly convergent, and therefore that the original series is +convergent; and it is obviously only one of a large class of series +possessing similar properties. + +\begin{Remark} +(ii) Let $\phi(x)$~be an increasing function such that $\phi(0) > 0$, $\phi \cgt x$. We +can define an increasing function~$f$, which satisfies the equation +\[ +\ff(x) = \phi(x), +\Tag{(1)} +\] +as follows. + +Draw the curves $y = x$, $y = \phi(x)$ (\Fig{5}). Take $Q_{0}$~arbitrarily on~$OP_{0}$ (see +the figure); draw~$Q_{0}R_{1}$ parallel +to~$OX$ and complete the rectangle~$Q_{0}Q_{1}$. +Join $Q_{0}$,~$Q_{1}$ by any +continuous arc everywhere inclined +at an acute angle to the +axes. On this arc take any +point~$Q$; draw $QP$,~$QR$ parallel +to the axes, and complete the +rectangle~$QQ'$. As $Q$~moves +from $Q_{0}$ to~$Q_{1}$, $Q'$~moves from +$Q_{1}$ to~$Q_{2}$, say. As we constructed +$Q'$ from~$Q$, so we can +construct $Q''$ from~$Q'$: proceeding +thus we define a continuous +curve $Q_{0}Q_{1}Q_{2}Q_{3}\dots$ corresponding +to a continuous and increasing +function~$f(x)$. Then +$f(x)$~satisfies~\Eq{(1)}. For if $y = f(x)$ +is the ordinate of~$Q$, it is clear that $\ff(x)$~is the ordinate of~$Q'$, which is equal +to~$\phi(x)$, the ordinate of~$P$. +%[Illustration: Fig. 5] +\Figure[0.6\textwidth]{5}{042} + +Let us write +\[ +f(x) = f_{1}(x), \qquad +\phi(x) = f_{1}f_{1}(x) = f_{2}(x), \qquad +f\phi(x) = \phi f(x) = f_{3}(x), +\] +and so on, so that $Q_{n}$~is the point $f_{n}(0)$,~$f_{n+1}(0)$. Also let $\psi$~be the function +inverse to~$\phi$, and write~$\psi_{2}$ for~$\psi\psi$, and so on. Finally, let the equation of~$Q_{0}Q_{1}$ +be $\theta(x, y) = 0$. Then it is easy to see that the equations of~$Q_{2n}Q_{2n+1}$ +and of~$Q_{2n+1}Q_{2n+2}$ are respectively +\[ +\theta\{\psi_{n}(x), \psi_{n}(y)\} = 0, \qquad +\theta\{\psi_{n+1}(y), \psi_{n}(x)\} = 0. +\] + +Suppose for example that $\phi(x) = e^{x}$, $OQ_{0} = a < 1$, and that $Q_{0}Q_{1}$~is the +straight line $y = a + \alpha x$, where $\alpha = (1 - a)/a$. Then the equations of~$Q_{2n}Q_{2n+1}$ +and of~$Q_{2n+1}Q_{2n+2}$ are +\[ +l_{n}y = a + \alpha l_{n}x, \qquad +l_{n}x = a + \alpha l_{n+1}y, +\] +or +\[ +y = e_{n-1}\{e^{\alpha} (l_{n-1}x)^{\alpha}\}, \qquad +y = e_{n}\{e^{-a/\alpha} (l_{n-1}x)^{1/\alpha}\}. +\] +%% -----File: 043.png---Folio 35------- +For simplicity let us take $a = \frac{1}{2}$, $\alpha = 1$. Then the equations of~$Q_{2n}Q_{2n+1}$ and +of~$Q_{2n+1}Q_{2n+2}$ are respectively +\begin{alignat*}{3} +y &= e_{n-1}\{\sqrt{e}(l_{n-1}x)\} &&= e_{n-2}\{(l_{n-2}x)^{\sqrt{e}}\} &&= \lambda_{n}(x),\\ +y &= e_{n}\{(l_{n-1}x)/ \sqrt{e}\} &&= e_{n-1}\{(l_{n-2}x)^{1/\sqrt{e}}\} &&= \mu_{n}(x), +\end{alignat*} +say. Now (\Ref{iv.}{§\;1}) +\[ +x^{\gamma} \clt \lambda_{3} \clt \dots \clt \lambda_{n} \clt \dots + \clt \mu_{n} \clt \dots \clt \mu_{3} \clt e^{x^{\gamma}} +\] +and a function~$f$, such that $\lambda_{n} \clt f \clt \mu_{n}$ for all values of~$n$, transcends the +logarithmico-exponential scales. But $f$~clearly satisfies these relations, and +so its increase is incapable of exact measurement by these scales. + +It is easily verified that $\lambda_{n}\lambda_{n}x \clt e^{x}$ and $\mu_{n}\mu_{n}x \cgt e^{x}$ for all values of~$n$. +Hence it is clear \textit{a~priori} that any increasing solution of~\Eq{(1)} must satisfy +$\lambda_{n} \clt f \clt \mu_{n}$. + +This kind of `graphical' method may also be employed to define functions +whose increase, like that of the function considered under (i) above, is slower +than that of any logarithm or more rapid than that of any exponential. It +can be employed, for example, to solve the equation +\[ +\phi(2^{x}) = 2\phi(x); +\] +and it can be proved that the increase of a function such that $\phi(2^{x}) \ceq \phi(x)$ +is slower than that of any logarithm (\Ref{vii.}{§\;3}). +\end{Remark} + +\Paragraph{6. The importance of the logarithmico-exponential scales.} +As we have seen in the earlier paragraphs of this section, it is possible, +in a variety of ways, to construct functions whose increase cannot be +measured by any $L$-function. It is none the less true that no one yet +has succeeded in defining a mode of increase genuinely independent of +all logarithmico-exponential modes. Our irregularly increasing functions +oscillate, according to a logarithmico-exponential law of oscillation, +between two logarithmico-exponential functions; the functions of~§\;5 +were constructed expressly to fill certain gaps in the logarithmico-exponential +scales. No function has yet presented itself in analysis +the laws of whose increase, in so far as they can be stated at all, cannot +be stated, so to say, in logarithmico-exponential terms. + +It would be natural to expect that the arithmetical functions which +occur in the theory of the distribution of primes might give rise to +genuinely new modes of increase. But, so far as analysis has gone, the +evidence is the other way. + +\begin{Remark} +Thus if we denote by~$\varpi(x)$ the number of prime numbers less than~$x$, it is +known that +\[ +\varpi(x) \sim \frac{x}{\log x}. +\] +%% -----File: 044.png---Folio 36------- + +More precisely +\[ +\varpi(x) = \int_{2}^{x} \frac{dt}{\log t} + \rho(x) = \Li(x) + \rho(x), +\] +where $|\rho(x)| \clt x(\log x)^{-\Delta}$. The precise order of~$\rho(x)$ has not yet been +determined, but there is reason to anticipate that $\rho(x) \cleq \sqrt{x}/(\log x)$. +\end{Remark} + + +\Chapter{VI.}{Differentiation and Integration.} + +\Paragraph{1. Integration.} It is important to know when relations of the +types $f(x) \cgt \phi(x)$, etc., can be differentiated or integrated. The +results are very much what might be expected from analogy with +similar results in other branches of analysis, and may therefore be +discussed somewhat summarily. For brevity we denote +\[ +\int_{a}^{x} f(t)\, dt, \qquad +\int_{a}^{x} \phi(t)\, dt +\] +(where $a$~is a constant) by $F(x)$ and~$\Phi(x)$. And we suppose for the +moment that $f$ and~$\phi$ are positive for $x \geqq a$. + +It may be well to repeat (cf.~\Ref{i.}{§\;4}) that $f$ and~$\phi$ are always supposed +to be (at any rate for $x > x_{0}$) positive, continuous, and monotonic, unless +the contrary is stated or clearly implied. Some of our conclusions are +valid under more general conditions; but the case thus defined, and +the corresponding case in which $f$ or~$\phi$ or~both of them are negative, +are the only cases of importance. + +\begin{Lemma} +If $\Phi \cgt 1$, and $f > H\phi$ for $x > x_{0}$, then $x_{1}$~can be found +so that $F > (H - \delta)\Phi$ for $x > x_{1}$: similarly $f < h\phi$ for $x > x_{0}$ involves +$F < (h + \delta)\Phi$ for $x > x_{1}$. +\end{Lemma} + +For if $f > H\phi$ for $x > x_{0}$, we have +\[ +F = \int_{a}^{x} f\, dt + > \int_{a}^{x_{0}} f\, dt + H \int_{x_{0}}^{x} \phi\, dt + > H\Phi + \int_{a}^{x_{0}} f\, dt - H \int_{a}^{x_{0}} \phi\, dt, +\] +and if we choose $x_{1}$ so that +\[ +\left(\int_{a}^{x_{0}} f\, dt + H \int_{a}^{x_{0}} \phi\, dt\right) \bigg/ \Phi < \epsilon +\] +for $x \geq x_{1}$, as we certainly can if $\Phi \cgt 1$, the result follows. Similarly +in the other case. From this lemma we can at once deduce the +following +%% -----File: 045.png---Folio 37------- + +\begin{Theorem} +Any one of the relations +\begin{alignat*}{5} +f &\cgt \phi, \qquad& +f &\clt \phi, \qquad& +f &\ceq \phi, \qquad& +f &\ceqq \phi, \qquad& +f &\sim \phi \\ +\intertext{involves the corresponding one of the relations} +F &\cgt \Phi, \qquad& +F &\clt \Phi, \qquad& +F &\ceq \Phi, \qquad& +F &\ceqq \Phi, \qquad& +F &\sim \Phi +\end{alignat*} +if either $F \cgt 1$ or $\Phi \cgt 1$. +\end{Theorem} + +To this we may add: \begin{Result}if both $\ds\int^{\infty} f\,dt$, $\ds\int^{\infty} \phi\,dt$ are convergent, then +$f \cgt \phi$, $f \clt \phi$, $f \ceq \phi$, $f \ceqq \phi$, $f \sim \phi$ involve corresponding relations between +\[ +\bar{F} = \int_{x}^{\infty} f\,dt, \qquad +\bar{\Phi} = \int_{x}^{\infty} \phi\,dt. +\] +\end{Result} + +The proof we may leave to the reader. These results have been +stated primarily for the case in which $f$~and~$\phi$ are positive; but there +is no difficulty in extending them to the case in which either function +or both are negative. + +\Paragraph{2. Differentiation.} {\Loosen It follows from~§\;1 that $f \cgt \phi$ involves +$f' \cgt \phi'$ if $f \cgt 1$ or $f \clt 1$ and \emph{if any one of the relations expressed by +$\cgt$,~$\clt$, $\ceq$,~$\ceqq$,~$\sim$ holds between $f'$~and~$\phi'$}.} + +\begin{Remark} +In interpreting this statement regard must be paid to the conventions +laid down in \Ref{i.}{§\;4}. Thus if $f \cgt \phi \cgt 1$, $f'$~and~$\phi'$ are positive; and $f' \cgt \phi'$. +But if $f \cgt 1 \cgt \phi$, $\phi$~is a decreasing function and $\phi' < 0$. In this case +$f' \cgt -\phi'$, a relation which we have agreed to denote by $f' \cgt \phi'$. If $1 \cgt f \cgt \phi$ +both $f'$~and~$\phi'$ are negative: the relation $-f' \clt -\phi'$ would involve +\[ +-\int_{x}^{\infty} f'\,dt \clt -\int_{x}^{\infty} \phi'\,dt +\] +or $f \clt \phi$, and is therefore impossible; similarly for $-f' \ceq -\phi'$; so we must +have $-f' \cgt -\phi'$, a relation which we have agreed also to denote by $f' \cgt \phi'$. +The case in which $f \ceq 1$ is exceptional; any one of the relations $f' \cgt \phi'$,~etc.\ +may then hold. Thus if $f = 1 + e^{-x}$, $f' = 1/x$, we have $f \cgt \phi$, $f' \clt \phi'$. The fact +is that in this case $f$, regarded as the integral of~$f'$, is dominated by the +constant of integration. +\end{Remark} + +Similar results hold, of course, for the relations $f \clt \phi$,~etc., with +similar exceptions. With regard to all of them it is to be observed +that the assumption that one of the relations holds between $f'$~and~$\phi'$ +is essential. We can never \emph{infer} that one of them holds. +We cannot even infer that $f'$~or~$\phi'$ is a steadily increasing or decreasing +function at all. Thus if $f = e^{x}$, $\phi = e^{x} + \sin e^{x}$, we have $f' = e^{x}$ and +$\phi' = e^{x}(1 + \cos e^{x})$. Thus $f$~and~$\phi$ increase steadily and $f \sim \phi$, $f' \sim f$; +%% -----File: 046.png---Folio 38------- +but $\phi'$~does not tend to infinity (vanishing for an infinity of values +of~$x$). Again if +\[ +\phi = e^{x}(\sqrt{2} + \sin x) + \tfrac{1}{2} x^{2}, +\] +we have +\[ +\phi' = e^{x} (\sqrt{2} + \sin x + \cos x) + x +\] +and $\phi \ceq e^{x}$, while $\phi'$~oscillates between the orders of $e^{x}$ and~$x$. It is +possible, though less easy, to obtain examples of this character in which +$\phi'$~also is monotonic. + +\Paragraph{3. Differentiation of $L$-functions.} If $f$~and~$\phi$ are $L$-functions, +so are $f'$~and~$\phi'$, and one of the relations $f' \cgt \phi'$, $f' \ceq \phi'$, $f' \clt \phi'$ +certainly holds (\Ref{iii.}{§\;2}). Thus in this case \emph{both differentiation and +integration are always legitimate}\footnotemark---this statement, however, being +subject to certain exceptions in the cases in which $f \ceq 1$ or $\phi \ceq 1$. +\footnotetext{A tacit assumption to this effect underlies much of Du~Bois-Reymond's work.} + +In what follows we shall suppose that all the functions concerned +are $L$-functions, or at any rate resemble $L$-functions in so far that one +of the relations $f \cgt \phi$, $f \ceqq \phi$, $f \clt \phi$ is bound to hold between any pair +of functions, and that differentiation and integration are permissible.\footnote + {The results which follow are all in substance due to Du~Bois-Reymond.} + +\begin{Result}[1.] If $f$~is an increasing function, and $f' \cgt f$, then $f \cgt e^{\Delta x}$. If +$f' \clt f$, then $f \clt e^{\delta x}$. Similarly if $f$~is a decreasing function, $f' \cgt f$ and +$f' \clt f$ involve $f \clt e^{-\Delta x}$ and $f \cgt e^{-\delta x}$ respectively. If $f' \ceqq f$, then +$e^{\delta x} \clt f \clt e^{\Delta x}$ or $e^{-\Delta x} \clt f \clt e^{-\delta x}$, and we can find a number~$\mu$ such +that $f = e^{\mu x} f_{1}$, where $e^{-\delta x} \clt f_{1} \clt e^{\delta x}$. +\end{Result} + +The proofs of these assertions are almost obvious. Thus if $f$~is an +increasing function, and $f' \cgt f$, we have +\[ +f'/f \cgt 1, \qquad +\log f \cgt x, +\] +and so $\log f > \Delta x$ for $x > x_{0}$, \ie\ $f > e^{\Delta x}$, or, what is the same thing, +$f \cgt e^{\Delta x}$. The last clause of the theorem follows at once from~\Ref{iii.}{§\;4}. + + +\begin{Result}[2.] More generally, if $v$~is any increasing function, $f'/f \cgt v'/v$ +involves $f \cgt v^{\Delta}$ or $f \clt v^{-\Delta}$, according as $f$~is an increasing or a decreasing +function; and $f'/f \clt v'/v$ involves $f \clt v^{\delta}$ or $f \cgt v^{-\delta}$. And $f'/f \ceqq v'/v$ +involves $v^{\delta} \clt f \clt v^{\Delta}$ or $v^{-\Delta} \clt f \clt v^{-\delta}$; and then we can find a number~$\mu$ +such that $f = v^{\mu}f_{1}$, where $v^{-\delta} \clt f_{1} \clt v^{\delta}$. +\end{Result} + +When $f$~is an increasing function we shall call $f'/f$ the \emph{type}~$t$ of~$f$:\footnote + {Du~Bois-Reymond calls $f/f'$ the type; the notation here adopted seems slightly + more convenient.} +it being understood that $t$~may be replaced by any simpler function~$\tau$ +such that $t \ceqq \tau$. The type of a \emph{decreasing} function~$f$ we define to be +%% -----File: 047.png---Folio 39------- +the same as that of the increasing function~$1/f$. The following table +shews the types of some standard functions: +\[ +\begin{array}{lcccccrlcc} +\text{\textit{Function}} & 1 & llx & lx & x^{\alpha} & e^{x} & e^{\alpha x^{\beta}} & e_{2}x & e_{3}x & \dots \\ +\text{\textit{Type}} & 0 & \dfrac{1}{x\, lx\, llx} & \dfrac{1}{x\, lx} & \dfrac{1}{x} & 1 & x^{\beta-1} & ex & e_{2}x\,ex & \dots +\end{array} +\] + +\begin{Remark} +If $f \cgt \phi$, then $f'/f \cgeq \phi'/\phi$. By making the increase of~$f$ large enough we +can make the increase of $t = f'/f$ as large as we please. The reader will find +it instructive to write out formal proofs of these propositions, and also of +the following. + +\Item{1.} As the increase of~$f$ becomes smaller and smaller, $f'/f$~tends to zero +more and more rapidly, but, so long as $f \to \infty$ at all, we cannot have +\[ +f'/f \clt \phi, \qquad +\int^{\infty} \phi\, dx \quad \text{\emph{convergent}}. +\] +On the other hand, if the last integral is divergent we can find~$f$ so that +$f \cgt 1$, $f'/f \clt \phi$. + +\Item{2.} Although we can find~$f$ so that $f'/f$~shall have an increase larger than +that of any given function of~$x$, we cannot have +\[ +f'/f \cgt \phi(f), \qquad +\int^{\infty} \frac{dx}{x\phi(x)} \quad \text{\emph{convergent}}. +\] +On the other hand, if the last integral is divergent we can find~$f$ so that +$f'/f \cgt \phi(f)$. + +{\Loosen[Thus we cannot find a function~$f$ which tends to infinity so slowly that +$f'/f \clt 1/x^{\alpha}$ ($\alpha > 1$). But we can find~$f$ so that $f'/f \clt 1 / x\, lx\, llx$ (\eg~$f = l_{3}x$). +We cannot find~$f$ so that $f'/f \cgt f^{\alpha}$ or $f' \cgt f^{1+\alpha}$ ($\alpha > 0$). But we can find~$f$ +so that $f'/f \cgt \lf$ (\eg~$f = e_{3}x$).]} + +\Item{3.} If $f \cgt e_{k}x$ for all values of~$k$, $f'/f$~satisfies the same condition, and +\[ +f' \cgt f\, \lf\, l_{2}f \dots l_{k}f. +\] + +He will also find it profitable to formulate corresponding theorems about +functions of a positive variable~$x$ which tends to zero. +\end{Remark} + +\Paragraph{4. Successive differentiation.} Du~Bois-Reymond has given +the following general theorem, which enables us to write down the +increase of any derivative of any logarithmico-exponential function. +We write $t$ for~$f'/f$, as in the last section, and we assume that no +derivative~$f^{(n)}$ satisfies $f^{(n)} \ceqq 1$: if this should be the case the results +of the theorem, so far as the derivatives $f^{(n+1)}$,~\dots\ are concerned, cease +to be true. + +\begin{Theorem} \Item{(i)} If $t \cgt 1/x$ \(so that $f \cgt x^{\Delta}$\) then +\[ +f \ceqq f'/t \ceqq f''/t^{2} \ceqq f'''/t^{3} \dots \ceqq f^{(n)}/t^{n} \dots. +\] +%% -----File: 048.png---Folio 40------- + +\Item{(ii)} If $t \clt 1/x$ \(so that $f \clt x^{\delta}$\) then +\[ +f \ceqq f'/t \ceqq xf''/t \ceqq x^{2}f'''/t \dots \ceqq x^{n-1} f^{(n)}/t \dots. +\] + +\Item{(iii)} If $t \ceqq 1/x$ \(so that $f = x^{\mu} f_{1}$, where $x^{-\delta} \clt f_{1} \clt x^{\delta}$\), then if $\mu$~is +not integral either set of formulae is valid. But if $\mu$~is integral +\[ +f \ceqq xf' \ceqq x^{2}f'' \dots \ceqq x^{\mu}f^{(\mu)} \ceqq x^{\mu} f^{(\mu+1)}/t_{1} \ceqq x^{\mu+1}f^{(\mu+2)}/t_{1} \dots, +\] +where $t_{1}$ is the type of~$f_{1}$. +\end{Theorem} + +\Item{(i)} If $t \cgt 1/x$, $1/t \clt x$ and so $t'/t^{2} \clt 1$; hence $t'/t \clt t = f'/f$ or +\[ +ft' \clt f't. +\] + +Differentiating the relation $f' \ceqq ft$, and using the relation just +established, we obtain +\[ +f'' \ceqq f't + ft' \ceqq f't. +\] + +Thus the type of~$f'$ is the same as that of~$f$; accordingly the +argument may be repeated and the first part of the theorem follows. + +\Item{(ii)} If $t \clt 1/x$, $xf' \clt f$ and so +\[ +xf'' + f' \clt f'. +\] + +But this cannot possibly be the case unless $xf'' \ceqq f'$. Differentiating +again we infer +\[ +xf''' + 2f'' \clt f'', +\] +whence $xf''' \ceqq f''$; and so on generally.\footnote + {More precisely $xf'' \sim -f'$, $xf''' \sim -2f''$, and so on.} +Thus the second part +follows. + +\Item{(iii)} If $t \ceq 1/x$, $f = x^{\mu}f_{1}$ and $t_{1}$,~the type of~$f_{1}$, satisfies $t_{1} \clt 1/x$. +Then +\[ +f' = \mu x^{\mu-1} f_{1} + x^{\mu}f_{1}' \ceqq x^{\mu-1} f_{1}(\mu + xt_{1}) \ceqq x^{\mu-1}f_{1}; +\] +Similarly $f'' \ceqq x^{\mu-2}f_{1}$ and so on. We can proceed indefinitely in this +way unless $\mu$~is integral: in this case we find $f^{(\mu)} \ceq f_{1}$, and from this +point we proceed as in case~(ii). + +\begin{Remark} +\textit{Examples.} \Item{(i)} If $f = e^{\sqrt{x}}$, then $t = 1/\sqrt{x} \cgt 1/x$, and $f^{(n)} \ceqq e^{\sqrt{x}}/(\sqrt{x})^{n}$. +If $f = e^{(\log x)^{2}}$, then $t = (\log x)/x \cgt 1/x$, and $f^{(n)} \ceqq e^{(\log x)^{2}} (\log x)^{n}/x^{n}$. + +\Item{(ii)} If $f = (\log x)^{m}$, then $t = 1/(x\log x) \clt 1/x$, and +\[ +f^{(n)} \ceqq tx^{-(n-1)}f \ceqq (\log x)^{m-1}/x^{n}. +\] + +\Item{(iii)} If $f = x^{2}\, llx$, $t \ceqq 1/x$. Here +\[ +f' \ceqq x\, llx, \qquad +f'' \ceqq llx, \qquad +f''' \ceqq 1/x\, lx, \qquad +f'''' \ceqq 1/x^{2}\, lx,\ \dots. +\] + +\Item{(iv)} The results of the theorem, in the first two cases, can be stated +more precisely as follows: + +If $t \cgt 1/x$, then +\[ +f^{(n)} \sim (f'/f)^{n}f. +\] +%% -----File: 049.png---Folio 41------- + +If $t \clt 1/x$, then +\[ +f^{(n)} \sim (-1)^{n-1} (n - 1)!\, x^{-(n-1)}f'. +\] + +If $f$~is a positive increasing function, then if $t \cgt 1/x$ all the derivatives are +ultimately positive, while if $t \clt 1/x$ they are alternately ultimately positive +and ultimately negative. +\end{Remark} + +\Paragraph{5. Functions of an integral variable.} The theorems for +functions of an integral variable~$n$, corresponding to those of §§\;1--4, +involve sums +\[ +A_{n} = a_{1} + a_{2} + \dots + a_{n} +\] +in place of integrals, and differences +\[ +\Delta a_{n} = a_{n} - a_{n+1} +\] +instead of differential coefficients. The reader will be able to +formulate and to prove for himself the theorems which correspond +to those of~§\;1. Thus +\begin{quote}`\begin{Result}% +$a_{n} \cgt b_{n}$, $a_{n} \clt b_{n}$, $a_{n} \ceq b_{n}$, $a_{n} \ceqq b_{n}$, $a_{n} \sim b_{n}$ involve the corresponding +equations for $A_{n}$,~$B_{n}$, if one at least of $A_{n}$,~$B_{n}$ tends +to infinity with~$n$% +\end{Result}' +\end{quote} +and so on.\footnote + {This is of course the well known theorem of Cauchy and Stolz: see Bromwich, + \textit{Infinite Series}, p.~377.} +Considerations of space forbid that we should go further +into the subject here. + + +\Chapter[Developments of the Infinitärcalcül.] +{VII.}{Some Developments of Du~Bois-Reymond's +Infinitärcalcül.} + +\Paragraph{1.} \First{We} shall conclude our account of the general theory by a brief +sketch of some interesting results due in the main to Du~Bois-Reymond. +For further details we must refer to his memoirs catalogued in the +Bibliographical Appendix. + +\Section{The functions $\dfrac{f(x + a)}{f(x)}$, $\dfrac{f(ax)}{f(x)}$, etc.} + +It is often necessary to obtain approximations to such functions as +\[ +f(x + a)/f(x), +\] +where $a$~is itself a function of~$x$, which for simplicity we suppose +positive, and which may tend to infinity with~$x$. In this connection +%% -----File: 050.png---Folio 42------- +Du Bois-Reymond\footnote + {\textit{Math.\ Annalen}, Bd.~8, S.~363 \textit{et~seq.}} +has proved a whole series of theorems: it will be +sufficient for our present purpose to give a few specimens of his results. +In what follows it will be assumed throughout that all the functions +dealt with are $L$-functions, or at any rate such that any pair of them +satisfy one of the relations $f \cgt \phi$, $f \ceqq \phi$, $f \clt \phi$, and that such +relations may be differentiated or integrated. This being so we +have +\[ +\frac{f(x + a)}{f(x)} = e^{\lf(x + \alpha) - \lf(x)} = e\left\{a\frac{f'(x + \alpha)}{f(x + \alpha)}\right\}, +\] +where $0 < \alpha < a$. This expression has certainly the limit unity if +$f' \cleq f$ and $a \clt 1$. Hence +\[ +f(x + a) \sim f(x) +\Tag{(1)} +\] +if $a \clt 1$ and $e^{-\Delta x} \clt f \clt e_{\Delta x}$. If $f'/f \clt 1$, \ie\ if $e^{-\delta x} \clt f \clt e^{\delta x}$, the +relation~\Eq{(1)} holds for $a \clt f/f'$: it certainly holds, for instance, if +$a = x\{f(x)\}^{-\mu}$, where $\mu > 0$, since $x/f^{\mu} \clt f/f'$ whenever $f \cgt 1$.\footnote + {For $\ds\int^{\infty} f^{-1-\mu} f'\,dx$ is convergent, and so $f'/f^{1+\mu} \clt 1/x$.} + +If $a \ceqq f/f'$ (as \eg\ if $f = e^{\mu x}f_{1}$, where $e^{-\delta x} \clt f_{1} \clt e^{\delta x}$, and $a \ceqq 1$), +$f(x + a)/f(x)$ will tend to a limit different from unity. + +Again +\[ +\frac{f(x + a)}{f(x)} = e\left\{a\frac{f'(x)}{f(x)}\, \frac{t(x + \alpha)}{t(x)}\right\}, +\] +where $t = f'/f$. Hence +\[ +\frac{f(x + a)}{f(x)} = e\left\{u\frac{f'(x)}{f(x)}\right\} \quad (u \sim a) +\Tag{(2)} +\] +{\Loosen in all cases in which $t(x + \alpha)/t(x) \sim 1$; as for example if $a \cleq 1$, +$e^{-\delta x} \clt t \clt e^{\delta x}$, or, what is the same thing, if} +\[ +a \cleq 1, \qquad +e^{-e^{\delta x}} \clt f \clt e^{e^{\delta x}}. +\] + +The reader will find it instructive to write down conditions under +which the equation~\Eq{(2)} holds when $u \ceqq a$ is substituted for $u \sim a$, and +to consider in what circumstances either relation holds when $a \cgt 1$. + +\Paragraph{2.} The reader is also recommended to verify some of the +following results: + +\begin{Remark} +\begin{Result} +\Item{(i)} If $1 \clt a \clt x$ and $x^{-\Delta} \clt f \clt x^{\Delta}$, then $f(x + a)/f(x) \sim 1$. + +\Item{(ii)} \Squeeze{If $f \clt x$ and $a \clt 1/f'$, or if $f \ceqq x$ and $a \clt 1$, then $f(x + a) - f(x) \clt 1$}. + +\Item{(iii)} If $e^{-\delta x} \clt f \clt e^{\delta x}$ and $a \clt f'/f''$, then +\[ +f(x + a) - f(x) \sim af'(x). +\] +\end{Result} +%% -----File: 051.png---Folio 43------- + +The condition $a \clt f'/f''$ may be simplified by means of the theorem of +\Ref{vi.}{§\;4}. Thus if $t \clt 1/x$ (\ie\ if $f \clt x^{\delta}$) it is equivalent to $a \clt x$. + +\begin{Result} +\Item{(iv)} If $x^{-\delta} \clt a \clt x^{\delta}$, $(lx)^{-\Delta} \clt f \clt (lx)^{\Delta}$, then $f(ax)/f(x) \sim 1$. + +\Item{(v)} If $e^{-\Delta\sqrt{lx}} \clt f \clt e^{\Delta\sqrt{lx}}$, then +\[ +\frac{f\{xf(x)\}}{f(x)} \ceqq 1, \qquad +e\left\{\frac{x\, \lf(x)f'(x)}{f(x)}\right\} \ceqq 1; +\] +and the limits of the two functions are the same: and if $e^{-\delta\sqrt{lx}} \clt e^{\delta\sqrt{lx}}$ this +limit is unity. +\end{Result} + +Suppose, \eg\ $f \cgt 1$, and let $f(x) = \phi(lx)$; then, if $a = f(x)$, +\[ +\frac{f(ax)}{f(x)} = e^{l\phi(lx + la) - l\phi(lx)} + = e^{la\phi'(lx + la_{1})/\phi(lx + la_{1})}, +\] +where $1 < a_{1} < a$. The exponent is +\[ +l\phi(lx + la_{1}) \frac{\phi'(lx + la_{1})}{\phi(lx + la_{1})}\, \frac{l\phi(lx)}{l\phi(lx + la_{1})}. +\] + +Now $a = f(x) \clt x^{\delta}$ and therefore $la_{1} \cleq la \clt lx$, and so, by~(i), +\[ +l\phi(lx + la_{1}) \sim l\phi(lx) +\] +if $l\phi \clt x^{\Delta}$ or if $f \clt e^{(lx)^{\Delta}}$, which is certainly the case. Hence the exponent +is asymptotically equivalent to +\[ +l\phi(u) \phi'(u)/\phi(u), +\] +where $u = lx + la_{1}$. And $l\phi(\phi'/\phi) \cleq 1$ if $(l\phi)^{2} \cleq u$, \ie\ if $\phi \cleq e^{\Delta\sqrt{u}}$ or +$f \cleq e^{\Delta\sqrt{lx}}$. In this case $f(ax) \ceqq f(x)$; and it is easy to see that if +$f \cleq e^{\delta\sqrt{lx}}$ the symbol~$\ceqq$ may be replaced by~$\sim$. + +\Item{(vi)} \emph{If $f(x) = x\phi(x)$, and $e^{-\delta\sqrt{lx}} \clt \phi \clt e^{\delta\sqrt{lx}}$, then} +\[ +f_{2}(x) \eqq \ff(x) \sim x\phi^{2},\ \dots,\ f_{n} \sim x\phi^{n},\ \dots. +\] + +The reader will easily prove this by the aid of the preceding results. He +will also find it instructive to calculate the increase of~$f_{n}$ when $f = e^{\sqrt{lx}}$ and +when $f = e^{(lx)^{\alpha}}$, where $\alpha > \frac{1}{2}$. +\end{Remark} + +\Section{The accuracy of approximations.} + +\Paragraph{3.} We have already (\Ref{iv.}{§§\;3--4}) had occasion to use the notion +of an approximation to the increase of a function, and to distinguish +legitimate and illegitimate forms of approximation. Du~Bois-Reymond +has given the following more precise definitions. + +He defines $\psi(x, u, u_{1}, \dots)$ to be an `approximate form' of~$y$ if +\[ +y = \psi(x, u, u_{1}, \dots), +\] +$\psi$~being a known function, and $u$,~$u_{1}$,~\dots\ unknown functions whose +increase is, however, subject to certain limitations. It is clear that +it is really useless, however, to insert more than one unknown function~$u$ +%% -----File: 052.png---Folio 44------- +in~$\psi$. The effect of the presence of~$u$ is to define a certain stretch +within which the increase of~$y$ lies, and the presence of several~$u$'s can +effect no more. We shall therefore consider only approximate forms +of the type +\[ +y = \psi(x, u). +\Tag{(1)} +\] + +Thus +\[ +e^{x^{u}} \quad (u \sim 1), \qquad +e^{(1+u)x} \quad (u \clt 1), \qquad +x^{1+u}e^{x} \quad (u \clt 1) +\Tag{(2)} +\] +are approximate forms of $y = xe^{x}/lx$; the second clearly closer than +the first and the third than the second. + +The closeness of an approximation may be measured as follows. +The presence of~$u$ in~\Eq{(1)} lends a certain degree of indeterminateness +to the increase of~$y$: all that we can say (the increase of~$u$ being +known to lie between certain limits) is that $y$~lies in a certain interval +\[ +\eta_{1} \cleq y \cleq \eta_{2}. +\] + +Now (\Ref{ii.}{§\;8}) we can find an increasing function~$F$ so that +$F(\eta_{1}) \ceq F(\eta_{2})$: if $F$~satisfies this condition, any more slowly increasing +function will do so too. \begin{Result}The slower the increase of~$F$ must +be taken, the rougher the approximation.\end{Result} + +{\Loosen The facts may be stated the other way round. Given~$y$, and a +function~$F$, such that $1 \clt F \clt x$, we can determine an interval +$\eta_{1} \cleq y \cleq \eta_{2}$ such that $F(\eta_{1}) \ceq F(\eta_{2})$. The slower the increase of~$F$, +the larger this interval may be taken; if $F \ceq x$ it vanishes, if $F \ceq 1$ +%[** TN: Hardy's notation for a closed interval; inconsistent, not modernizing] +it may be taken as large as we please. If $F = lx$ it might be $(y^{\delta}, y^{\Delta})$; +if $F = l_{2}x$ it might be} +\[ +e^{(ly)^{\delta}}, \qquad +e^{(ly)^{\Delta}}, +\] +and so on. No logarithmico-exponential form of~$F$, however, can give +an interval as large as~$(\log y, e^{y})$; a function~$F$ such that $F(y) \ceq F(e^{y})$ +must transcend any logarithmico-exponential scale. + +\begin{Remark} +Let us consider the approximations~\Eq{(2)} for~$xe^{x}/lx$. + +\Item{(i)} If $y = e^{x^{u}}$ ($u\sim l$), $y$~lies in the interval $e^{x^{1-\delta}}$, $e^{x^{1+\delta}}$. Since +\[ +ll(e^{x^{1-\delta}}) = (1 - \delta)lx \ceq ll(e^{x^{1+\delta}}) +\] +we may take $F = llx$, or even $F = (llx)^{\Delta}$: but the increase of~$F$ cannot be +taken as large as~$(lx)^{\delta}$. + +\Item{(ii)} If $y = e^{(1+u)x}$ ($u \clt 1$), $y$~lies in the interval $e^{(1-\delta)x}$, $e^{(1+\delta)x}$. Then we +may take $F = (lx)^{\Delta}$, but we cannot take $F = e^{(lx)^{\delta}}$. + +\Item{(iii)} {\Loosen If $y = x^{1+u}e^{x}$ we may, as the reader will easily verify, take $F = e^{(lx)^{\mu}}$, +where $\mu$~is any number less than unity.} +%% -----File: 053.png---Folio 45------- + +Another example of an approximation is given by the formula +\[ +\frac{f(x + a)}{f(x)} = e\left\{u\frac{f'(x)}{f(x)}\right\} \quad (u \sim a). +\] + +If, \eg, $a$~is a constant, +\[ +l\left\{\frac{f(x + a)}{f(x)}\right\} + \sim l\left\{e\left[\frac{f'(x)}{f(x)}\right]\right\}, +\] +and the degree of accuracy of the approximation is great enough to be +measured by the function $F = lx$. +\end{Remark} + +\Section{The approximate solution of equations.} + +\Paragraph{4.} It is often important to obtain an asymptotic solution of an +equation $f(x, y) = 0$, \ie\ to find a function whose increase gives an +approximation to that of~$y$. No very general methods of procedure +can be given, but the kind of methods which may be pursued are +worth illustrating by a few examples. + +\Item{(i)} Suppose that the equation is +\[ +x = y\kappa(y), +\Tag{(1)} +\] +where $y^{-\delta} \clt \kappa \clt y^{\delta}$. If the increase of~$\kappa$ is so slow that $\kappa\{y\kappa(y)\} \ceq \kappa(y)$ +it is clear that +\[ +y \ceq x/\kappa(y) \ceq x/\kappa(x): +\] +and if the increase of~$\kappa$ is slow enough we may have $y \sim x/\kappa(x)$. + +The conditions +\[ +e^{-\Delta\sqrt{ly}} \clt \kappa(y) \clt e^{\Delta\sqrt{ly}}, \qquad +e^{-\delta\sqrt{ly}} \clt \kappa(y) \clt e^{\delta\sqrt{ly}} +\] +are, by the result~(v) of~§\;2, enough to ensure the truth of these +hypotheses; and then $y = ux/\kappa(x)$, where $u \ceq 1$ (or $u \sim 1$) is an +approximate solution of our equation. + +\begin{Remark} +Du~Bois-Reymond has proved that the more elaborate approximations +\[ +y = ux/\{\kappa(x/\kappa)\}, \qquad +y = ux\kappa^{-1/\{1+(x\kappa'/\kappa)\}} +\] +have a wider range of validity: and that more elaborate approximations still +may be constructed valid within the range +\[ +e^{-\Delta(ly)^{1-\delta}} \clt \kappa \clt e^{\Delta (ly)^{1-\delta}}. +\] +\end{Remark} + +The more general equation +\[ +x = y^{m}\kappa(y) +\] +can clearly be reduced to the form considered above by writing~$x^{m}$ for~$x$ +and $\kappa^{m}$ for~$\kappa$. +%% -----File: 054.png---Folio 46------- + +In general, if $x = \phi(y)$, the more rapid the increase of~$\phi$ the more +precisely can we determine the increase of~$y$ as a function of~$x$. Thus if +\[ +x = ye^{y} +\] +we have $lx = y + ly$ and +\[ +y = lx - ly = lx(1 + u), +\] +where $u \sim ly/lx \sim llx/lx$. This is a solution of a much more precise +kind than those considered above. + +\Paragraph{5.} The reader will find it instructive to examine the following +results: + +\begin{Remark} +\Item{(i)} Let +\[ +x = ye^{(ly)^{3/8}}. +\] + +This is an example of the work of~§\;4: and +\[ +y \sim xe^{-(lx)^{3/8}}. +\] + +\Item{(ii)} Let +\[ +x = ye^{(ly)^{5/8}}. +\] + +Here +\begin{align*} +y &\sim xe [-(lx)^{5/8} \{1-(lx)^{-3/8}\}^{5/8}]\\ + &\sim xe\{-(lx)^{5/8} + \tfrac{5}{8}(lx)^{1/4}\}. +\end{align*} + +\Item{(iii)} Let +\[ +x = y^{m}(ly)^{m_{1}}(l_{2}y)^{m_{2}} \dots (l_{r}y)^{m_{r}}. +\] + +Here +\[ +y \sim m^{m_{1}/m} x^{1/m} (lx)^{-m_{1}/m} \dots (l_{r}x)^{-m_{r}/m}. +\] + +\Item{(iv)} Let +\[ +x = e^{y^{2}}ly. +\] + +Here +\[ +y = \sqrt{lx - l_{3}x} + u \quad (u \clt 1). +\] + +\Item{(v)} As an example of another type, Du~Bois-Reymond has considered +the equation +\[ +f(x + y) - f(x) = C, +\] +where $C$~is a positive constant. He finds +\begin{gather*} +y \sim C/f'(x) \quad (f(x) \cgt lx),\\ +y = xe\{Cu/xf'(x)\} \quad (u \sim 1,\ lx \cgt f(x) \cgt llx), +\end{gather*} +and so on: the forms of the solution when $f \ceq lx$, $f \ceq llx$,~\dots\ are exceptional. + +\Item{(vi)} As an example of an approximation pushed to greater lengths let us +take the following result: if +\[ +x = y\, ly, +\] +then +\[ +y = \frac{x}{lx} \left\{1 + \frac{llx}{lx} + \frac{(llx)^{2}}{(lx)^{2}} - \frac{llx}{(lx)^{2}}\right\} + u, +\] +where +\[ +u \ceqq \frac{x(llx)^{3}}{(lx)^{4}}. +\] +\end{Remark} + +\Paragraph{6.} Here we may bring our account of the general theory to a +close. It is a theory that has found, and is finding, a large and +increasing variety of applications in various branches of mathematics: +the nature of some of these applications the reader may glean from +Appendix~II\@. +%% -----File: 055.png---Folio 47------- + + +\Appendix{I.}{General Bibliography.} + +\Author{Du~Bois-Reymond}'s memoirs bearing on the subjects of this tract are: + +\Work Sur la grandeur relative des infinis des fonctions (\textit{Annali di +Matematica}, Serie~2, t.~4, p.~338). + +\Work Théorème général concernant la grandeur relative des infinis +des fonctions et de leurs derivées (\textit{Crelle's Journal}, Bd.~74, S.~294). + +\Work Eine neue Theorie der Convergenz und Divergenz von Reihen +mit positiven Gliedern. \textit{Anhang}: Ueber die Tragweite der +logarithmischen Kriterien (\textit{Crelle's Journal}, Bd.~76, S.~61). + +\Work Ueber asymptotische Werthe, infinitäre Approximationen, und +infinitäre Auflösung von Gleichungen (\textit{Math.\ Annalen}, Bd.~8, +S.~363). Nachtrag zur vorstehenden Abhandlung (\textit{ibid.}, S.~574). + +\Work Notiz über infinitäre Gleichheiten (\textit{Math.\ Annalen}, Bd.~10, +S.~576). + +\Work Ueber die Paradoxen des Infinitärcalcüls (\textit{Math.\ Annalen}, +Bd.~11, S.~149). + +\Work Notiz über Convergenz von Integralen mit nicht verschwindendem +Argument (\textit{Math.\ Annalen}, Bd.~13, S.~251). + +\Work Ueber Integration und Differentiation infinitären Relationen +(\textit{Math.\ Annalen}, Bd.~14, S.~498). + +\Work Ueber den Satz: $\lim f'(x) = \lim f(x)/x$ (\textit{Math.\ Annalen}, +Bd.~16, S.~550). + +See also + +\Author{A. Pringsheim}: Ueber die sogenannte Grenze und die Grenzgebiete +zwischen Convergenz und Divergenz (\textit{Münchener Sitzungsberichte}, +Bd.~26, S.~605). + +\Same Ueber die Du~Bois-Reymond'sche Convergenz-Grenze u.s.w. +(\textit{Münchener Sitzungsberichte}, Bd.~27, S.~303). + +\Same Allgemeine Theorie der Convergenz und Divergenz von Reihen +mit positiven Gliedern (\textit{Math.\ Annalen}, Bd.~35, S.~347). + +\Same Zur Theorie der bestimmten Integrale und der unendlichen +Reihen (\textit{Math.\ Annalen}, Bd.~37, S.~591). + +\Author{J. Hadamard}: Sur les caractères de convergence des séries à termes +positifs et sur les fonctions indéfiniment croissantes (\textit{Acta +Mathematica}, t.~18, p.~319 and p.~421). +%% -----File: 056.png---Folio 48------- + +\Author{S. Pincherle}: Alcune osservazioni sugli ordini d'infinito delle funzioni +(\textit{Memorie della Accademia delle Scienze di Bologna}, Ser.~4, t.~5, +p.~739). + +\Author{E. Borel}: \textit{Leçons sur les fonctions entières}, pp.~111--122. + +\Same \textit{Leçons sur les séries à termes positifs}, pp.~1--50. + +\Same \textit{Leçons sur la théorie de la croissance.} + + +\Appendix[Applications.] +{II.}{A Sketch of Some Applications,\protect\footnotemark\ With References.} +\footnotetext{That is to say of certain regions of mathematical theory in which the notation + and the ideas of the \textit{Infinitärcalcül} may be used systematically with a great gain + in clearness and simplicity.} + +\Section[A.]{Convergence and divergence of series and integrals.} + +\Subsection{(i)}{The logarithmic tests.} The series $\sum u_{n}$ ($u_{n} \geq 0$) is convergent if +\begin{flalign*} +&&u_{n} &\cleq n^{-1-\alpha} && \\ +\RTag{\emph{or}} +&&u_{n} &\cleq (n\, ln \dots l_{k-1}n)^{-1}(l_{k}n)^{-1-\alpha}, && +\end{flalign*} +where $\alpha > 0$, and divergent if +\begin{flalign*} +&&u_{n} &\cleq n^{-1} && \\ +\RTag{\emph{or}} +&&u_{n} &\cgeq (n\, ln \dots l_{k}n)^{-1}(l_{k}n)^{-1}. && +\end{flalign*} + +The integral $\ds\int^{\infty} f(x)\,dx$ ($f \geqq 0$) is convergent if +\begin{flalign*} +&&f &\cleq x^{-1-\alpha} && \\ +\RTag{\emph{or}} +&&f &\cleq (x\, lx \dots l_{k-1}x)^{-1}(l_{k}x)^{-1-\alpha}, && +\end{flalign*} +where $\alpha > 0$, and divergent if +\begin{flalign*} +&&f &\cleq x^{-1} && \\ +\RTag{\emph{or}} +&&f &\cleq (x\, lx \dots l_{k}x)^{-1}. && +\end{flalign*} + +The integral $\ds\int_{0} f(x)\,dx$ ($f \geqq 0$) is convergent if +\begin{flalign*} +&&f &\cleq (1/x)^{1-\alpha} && \\ +\qquad{\emph{or}} +&&f &\cleq (1/x)\{l(1/x) \dots l_{k-1}(1/x)\}^{-1} \{l_{k}(1/x)\}^{-1-\alpha}, && +\end{flalign*} +where $\alpha > 0$, and divergent if +\begin{flalign*} +&&f &\cgeq 1/x && \\ +\qquad{\emph{or}} +&&f &\cgeq (1/x)\{l(1/x) \dots l_{k}(1/x)\}^{-1}. && +\end{flalign*} +%% -----File: 057.png---Folio 49------- + +[The first general statement of the `logarithmic criteria,' so far +as series are concerned, appears to have been made by De~Morgan: +see his \textit{Differential and Integral Calculus}, 1839, p.~326. The +essentials of the matter, however, appear in a posthumous memoir +of Abel (\textit{\OE uvres complètes}, t.~2, p.~200; see also t.~1, p.~399). This +memoir appears also to have been first published in 1839. The case +of $k = 1$ had been dealt with by Cauchy (\textit{Exercices de Mathématiques}, +t.~2, 1827, pp.~221 \textit{et~seq.}). Bertrand appears to have arrived at +some or all of De~Morgan's results independently (see \textit{Liouville's Journal}, +t.~7, 1842, p.~37) and the criteria are very commonly attributed to him. +The criteria for integrals do not appear to have been stated generally +before Riemann, \textit{Inaugural-Dissertation} of 1854 (\textit{Werke}, S.~229). + +The following references may also be useful: + +Bonnet, \textit{Liouville's Journal}, t.~8, p.~78. + +Dini, \textit{Sulle serie a termini positivi} (Pisa, 1867); also in the +\textit{Annali dell' Univ.\ Tosc.}, t.~9, p.~41. + +Du~Bois-Reymond, \textit{Crelle's Journal}, Bd.~76, S.~619. + +Pringsheim, \textit{Math.\ Annalen}, Bd.~35, S.~347 and Bd.~37, S.~591; +also in the \textit{Encyklopädie der Math.\ Wiss.}, Bd.~1, Th.~1, S.~77 \textit{et~seq.} + +Hobson, \textit{Theory of functions of a real variable}, p.~406. + +Bromwich, \textit{Infinite series}, pp.~29,~37. + +Hardy, \textit{Course of pure mathematics}, pp.~357 \textit{et~seq.} + +Chrystal, \textit{Algebra}, vol.~2, pp.~109 \textit{et~seq.}] + +\Subsection{(ii)}{General theorems analogous to Du~Bois-Reymond's Theorem +\(\Ref{ii.}{§\;1}\).} + +Given any divergent series $\sum u_{n}$ of positive terms, we can find a +function~$v_{n}$ such that $v_{n} \clt u_{n}$ and $\sum v_{n}$ is divergent; \ie\ given any +divergent series we can find one more slowly divergent. + +Given any convergent series $\sum u_{n}$ of positive terms we can find~$v_{n}$ +so that $v_{n} \cgt u_{n}$ and $\sum v_{n}$ is convergent; \ie\ given any convergent +series we can find one more slowly convergent. + +Given any function~$\phi(n)$ tending to infinity, however slowly, we +can find a convergent series~$\sum u_{n}$ and a divergent series~$\sum v_{n}$ such +that $v_{n}/u_{n} = \phi(n)$. + +Given an infinite sequence of series, each converging (diverging) +more slowly than its predecessor, we can find a series which converges +(diverges) more slowly than any of them. + +[See Abel and Dini, \lc~\textit{supra}; Hadamard, \textit{Acta Mathematica}, t.~18, +p.~319 and t.~27, p.~177; Bromwich, \textit{Infinite series}, p.~40; Littlewood, +\textit{Messenger of Mathematics}, vol.~39, p.~191.] +%% -----File: 058.png---Folio 50------- + +There is no function~$\phi(n)$ such that $u_{n}\phi(n) \cgeq 1$ is a necessary +condition for the divergence of $\sum u_{n}$, and no function~$\phi(n)$ such that +$\phi(n) \cgt 1$ and $u_{n}\phi(n) \cleq 1$ is a necessary condition for the convergence +of~$\sum u_{n}$. + +If $u_{n}$~is a \emph{steadily decreasing} function of~$n$, then $nu_{n} \clt 1$ \emph{is} a +necessary condition for convergence; but there is no function~$\phi(n)$ +such that $\phi(n) \cgt 1$ and $n\phi(n) u_{n} \clt 1$ is a necessary condition. + +[Pringsheim, \textit{Math.\ Annalen}, Bd.~35, S.~343 \textit{et~seq.; ibid.}, Bd.~37, +S.~591 \textit{et~seq.}] + +If however $nu_{n}$~decreases steadily, then $n\log nu_{n} \to 0$ is a necessary +condition; and if $n\psi(n)u_{n}$, where $n\psi(n) \cgt 1$ and $\ds\int \frac{dn}{n\psi(n)} \cgt 1$, decreases +steadily, then +\[ +\left(n\psi(n) \int \frac{dn}{n\psi(n)}\right) u_{n} \to 0 +\] +is a necessary condition. + +\Subsection{(iii)}{Special series and integrals possessing peculiarities in respect +to the mode of their convergence or divergence.} + +For examples of series and integrals which converge or diverge +so slowly as not to answer to any of the logarithmic criteria see +Du~Bois-Reymond, Pringsheim, Borel (\lc~\textit{supra}), and Blumenthal, +\textit{Principes de la théorie des fonctions entières d'ordre infini}, ch.~1. + +In these cases the logarithmic tests are insufficient (\textit{en~défaut}, +\Ref{iv.}{§§\;2,~5}). For examples of series and integrals to which the +logarithmic tests are \emph{inapplicable} (\Ref{v.}{§§\;3,~4}) see the writings just +mentioned and also + +Thomae: \textit{Zeitschrift für Mathematik}, Bd.~23, S.~68. + +Gilbert: \textit{Bulletin des Sciences Mathématiques}, t.~12, p.~66. + +Goursat: \textit{Cours d'Analyse}, t.~1, p.~205. + +Hardy: \textit{Messenger of Mathematics}, vol.~31, p.~1; \textit{ibid.},~vol.~31, +p.~177; \textit{ibid.},~vol.~39, p.~28. + +\Section[B.]{Asymptotic formulae for finite series and integrals.} + +A closely connected problem is that of the determination of +asymptotic formulae for +\[ +A_{n} = a_{1} + a_{2} + \dots + a_{n} +\] +or for +\[ +\Phi(x) = \int_{a}^{x} \phi(t)\,dt, +\] +{\Loosen when the behaviour of $a_{n}$ or~$\phi(x)$ for large values of $n$ or~$x$ is known. +A good deal can be accomplished in this direction by means of +%% -----File: 059.png---Folio 51------- +(i)~the theorem of Cauchy and Stolz, that, if $a_{n}$~and~$b_{n}$ are positive and +$a_{n} \sim Cb_{n}$, then $A_{n} \sim CB_{n}$, (ii)~the theorems of~\Ref{vi.}{}\ and (iii)~the theorem +of Maclaurin and Cauchy, that} +\[ +\phi(1) + \phi(2) + \dots + \phi(n) - \int_{1}^{n} \phi(x)\,dx, +\] +where $\phi(x)$~is a positive and decreasing function of~$x$, tends to a limit +as $n \to \infty$. + +[For~(i) see Cauchy, \textit{Analyse algébrique}, p.~52; Stolz, \textit{Math.\ +Annalen}, Bd.~14, S.~232, or \textit{Allgemeine Arithmetik}, Bd.~1, S.~173; +Jensen, \textit{Tidskrift for Mathematik}~(5), Bd.~2, S.~81; Bromwich, +\textit{Infinite series}, p.~378, and \textit{Proc.\ Lond.\ Math.\ Soc.}, ser.~2, vol.~7, +p.~101. Proofs of~(iii) will be found in almost any modern treatise +on analysis: \eg, Bromwich, \textit{Infinite series}, p.~29; Hardy, \textit{Course +of pure mathematics}, p.~305. An important extension to \emph{slowly +oscillating} series has been given recently by Bromwich (\textit{Proc.\ Lond.\ +Math.\ Soc.}, ser.~2, vol.~7, p.~327).] + +Among the most important results which follow from these +theorems are +\begin{gather*} +1^{s} + 2^{s} + \dots + n^{s} \sim \frac{n^{s+1}}{s + 1} \quad (s > -1), \\ +1^{s} + 2^{s} + \dots + n^{s} - \frac{n^{s+1}}{s + 1} \sim \zeta(-s) \quad (-1 < s < 0), \\ +1 + \frac{1}{2} + \dots + \frac{1}{n} - \log n \sim \gamma, +\end{gather*} +\begin{multline*} +1 + \frac{\alpha·\beta}{1·\gamma} + + \frac{\alpha(\alpha + 1) \beta(\beta + 1)}{1·2·\gamma(\gamma + 1)} + \dots\ + \text{to $n$~terms,} \\ +\begin{aligned} + &\sim \frac{\Gamma(\gamma)}{\Gamma(\alpha)\, \Gamma(\beta)}\, + \frac{n^{\alpha+\beta-\gamma}}{\alpha + \beta - \gamma}\quad + (\alpha + \beta > \gamma), \\ +\LTag{\emph{or}} + &\sim \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\, \Gamma(\beta)}\, + \log n\quad + (\alpha + \beta = \gamma). +\end{aligned} +\end{multline*} + +In connection with the last result see Bromwich, \textit{Proc.\ Lond.\ Math.\ +Soc.}, ser.~2, vol.~7, p.~101; in the earlier formulae $\gamma$~is Euler's constant +and $\zeta$~denotes the `Riemann $\zeta$-function.' + +The most important of all formulae of this kind is beyond question +\[ +\log 1 + \log 2 + \dots + \log n - (n + \tfrac{1}{2})\log n + n \sim \tfrac{1}{2} \log(2\pi), +\] +which, in the form +\[ +n! \sim n^{n+\frac{1}{2}} e^{-n} \sqrt{2\pi}, +\] +constitutes \emph{Stirling's Theorem}. The literature connected with Stirling's +Theorem and its extensions to the Gamma-function of a non-integral +%% -----File: 060.png---Folio 52------- +or complex variable is far too extensive to be summarized here. See +\textit{Encykl.\ der Math.\ Wiss.}, Bd.~II.~(2), S.~165 \textit{et~seq.}; Bromwich, \textit{Infinite +series}, pp.~461 \textit{et~seq.} + +Another formula of the same kind is +\[ +1^{1}2^{2}3^{3} \dots n^{n} \sim An^{\frac{1}{2}n^{2} + \frac{1}{2}n + \frac{1}{12}}\, e^{-\frac{1}{4}n^{2}}, +\] +where $A$~is a constant defined by the equation +\[ +\log A = \tfrac{1}{12}\log 2\pi + \tfrac{1}{12} \gamma + + \frac{1}{2\pi^{2}} \sum_{1}^{\infty} \frac{\log \nu}{\nu^{2}}. +\] + +The properties of this constant have been investigated by Kinkelin +and Glaisher (Kinkelin, \textit{Crelle's Journal}, Bd.~57, S.~122: Glaisher, +\textit{Messenger of Mathematics}, vol.~6, p.~71; vol.~7, p.~43; vol.~23, p.~145; +vol.~24, p.~1; \textit{Quarterly Journal of Mathematics}, vol.~26, p.~1: see also +Barnes, \textit{ibid.},~vol.~31, pp.~264 \textit{et~seq.}). + +All these results are intimately bound up with the theory of +the general `Euler-Maclaurin Sum Formula' +\[ +\sum_{1}^{n} f(n) + = \int^{n} f(x)\,dx + C + \tfrac{1}{2}f(n) + \frac{B_{1}}{2!} f'(n) - \frac{B_{2}}{4!} f'''(n) + \dots +\] +which also possesses an extensive literature (see Schlömilch, \textit{Theorie +der Differenzen und Summen}; Boole, \textit{Finite differences}; Markoff, +\textit{Differenzenrechnung}; Seliwanoff, \textit{Differenzenrechnung}; \textit{Encykl.\ der +Math.\ Wiss.}, Bd.~I. S.~929 \textit{et~seq.}; Bromwich, \textit{Infinite series}, +p.~238 and p.~324; Barnes, \textit{Proc.\ Lond.\ Math.\ Soc.}, ser.~2, vol.~3, +pp.~253 \textit{et~seq.}; where many further references are given). + +A simple example of the use of the general formula is afforded +by the relation +\[ +\sum_{1}^{n} \nu^{s} - \frac{n^{s+1}}{s + 1} - \tfrac{1}{2} n^{s} + - \sum_{1} (-1)^{i-1} \left(\frac{s}{2i - 1}\right) \frac{B_{i}}{2i} n^{s-2i+1} \sim \zeta(-s). +\] + +Here $s$~is positive and not integral, and the summation with +respect to~$i$ is continued until we come to a negative power of~$n$. + +\Section[C.]{Formulae involving prime numbers only.} + +Asymptotic formulae involving functions defined arithmetically, +and particularly functions defined by sums of functions of prime +numbers only, play a most important part in the analytical theory +of numbers. Of these the most important is the formula +\[ +\Pi(n) \sim \frac{n}{ln}, +\] +where $\Pi(n)$~denotes the number of prime numbers less than~$n$. +%% -----File: 061.png---Folio 53------- + +Similarly it is known that +\[ +\sum lp \sim n, \qquad +\sum \frac{lp}{p} \sim ln, \qquad +\sum \frac{1}{p} \sim lln +\] +(the summation in each case applying to all primes less than~$n$) while +$\sum\limits^{\infty} \dfrac{1}{p\, lp}$ is convergent. + +Many more accurate results have been established by recent +writers, particularly Mertens, Hadamard, Von~Mangoldt, De~la~Vallée-Poussin, +and Landau; and the theory has to a considerable extent +been freed from Riemann's still unproved assumption that all the +roots of his Zeta-function have their real part equal to~$\frac{1}{2}$. Thus it +has been shown that +\[ +\Pi(n) = \int_{2}^{n} \frac{dx}{\log x} + O\left\{\frac{n}{(ln)^{\Delta}}\right\}, +\] +or, still more accurately, +\[ +\Pi(n) = \int_{2}^{n} \frac{dx}{\log x} + O\{ne^{-\alpha\sqrt{ln}}\}, +\] +where $\alpha$~is a positive constant; but it still remains to be settled +whether (as there is some reason to suppose) the last term can be +replaced by~$O(\sqrt{n})$ or even by +\[ +O\left(\frac{\sqrt{n}}{ln}\right). +\] + +[It would carry us too far to give detailed references to the +literature of this exceedingly difficult and fascinating subject. The +reader should consult Landau's exhaustive \textit{Handbuch der Lehre von +der Verteilung der Primzahlen} (Teubner, 1909).] + +\Section[D.]{The theory of integral functions.} + +\Subsection{1.}{}The series $\sum c_{n}x^{n}$ will converge for all values of~$x$ (real or +complex), and so define an \emph{integral function}~$f(x)$, if and only if +$\sqrt[n]{|c_{n}|} \to 0$, \ie\ if $|c_{n}| \clt e^{-\Delta n}$. + +\Subsection{2.}{The three indices of a function of finite order.} The three +most important characters of an integral function~$f(x)$ are: + +\Item{(i)} $\gamma_{n} = |c_{n}|$, the modulus of the $n$th~coefficient; + +\Item{(ii)} $\alpha_{n} = |a_{n}|$, the modulus of the $n$th (in order of absolute +magnitude) zero of~$f(x)$; + +\Item{(iii)} $M(r)$, the maximum of~$|f(x)|$ on the circle $|x| = r$. $M(r)$~is +known to be an increasing function of~$r$, and in all cases $M(r) \cgt r^{\Delta}$. +%% -----File: 062.png---Folio 54------- + +A function such that $M(r) \clt e^{r^{\Delta}}$ is called a \emph{function of finite +order}. We shall confine our remarks to such functions. + +The principal problem of the theory of integral functions is the +determination of the relations between the increases of $\alpha_{n}$,~$1/\gamma_{n}$, and~$M(r)$. +Those which subsist between the two latter functions are the +simplest: when $\alpha_{n}$~is taken into account the theory is complicated by +the `Picard case of exception'---the case of functions which (like~$e^{x}$) +have no zeroes, or whose zeroes are scattered abnormally widely over +the plane. + +The nature of the results of the general theory may be gathered +from a statement of a few of the simplest of them. + +If +\[ +n^{-\mu-\delta} \clt \sqrt[n]{\gamma_{n}} \clt n^{-\mu+\delta}, +\] +\ie\ if +\[ +l(1/\gamma_{n}) \sim \mu n\, ln, +\] +we call $\mu$ the \emph{$\mu$-index}. The index may be defined in \emph{all} cases without +any assumption as to the existence of a limit for $\{l(1/\gamma_{n})/(n\, ln)\}$; we +confine ourselves to the simplest case. + +If +\[ +n^{(1/\lambda)-\delta} \clt \alpha_{n} \clt n^{(1/\lambda)+\delta}, +\] +we call $\lambda$ the \emph{$\lambda$-index}; and if +\[ +e^{r^{\nu-\delta}} \clt M(r) \clt e^{r^{\nu+\delta}}, +\] +we call $\nu$ the \emph{$\nu$-index}: thus +\[ +l\alpha_{n} \sim (ln)/\lambda, \qquad +llM(r) \sim \nu\, lr. +\] + +Then $\mu = 1/\nu$: and \emph{in general} $\lambda = \nu$. + +Thus for the function +\[ +\frac{\sin(\sqrt{x})}{\sqrt{x}} = 1 - \frac{x}{3!} + \frac{x^{2}}{5!} - \dots +\] +we have $\lambda = \nu = \frac{1}{2}$ and $\mu = 2$, as the reader will easily verify (using +Stirling's Theorem to determine~$\mu$). + +\Subsection{3.}{Special results.} More precise results than these have been +obtained in many cases. Thus if +\[ +\{n(ln)^{-\alpha_{1}} \dots (l_{\nu}n)^{-\alpha_{\nu}+\delta}\}^{-1/\rho} + \clt \sqrt[n]{\gamma_{n}} + \clt \{n(ln)^{-\alpha_{1}} \dots (l_{\nu}n)^{-\alpha_{\nu}-\delta}\}^{-1/\rho}, +\] +then +\[ +e\{r^{\rho}(lr)^{\alpha_{1}} \dots (l_{\nu}r)^{\alpha_{\nu}-\delta}\} + \clt M(r) + \clt e\{r^{\rho}(lr)^{\alpha_{1}} \dots (l_{\nu}r)^{\alpha_{\nu}+\delta}\}, +\] +and conversely. +%% -----File: 063.png---Folio 55------- + +As examples of still more accurate, but more special results, we +may quote the following: +\begin{align*} +&\sum \frac{x^{n}}{n^{\alpha n}} + \sim \sqrt{\frac{2\pi}{e\alpha}}\, x^{1/2\alpha} e^{(\alpha/e)x^{1/\alpha}},\\ +&\sum \frac{x^{n}}{(n!)^{\alpha}} + \sim \frac{1}{\sqrt{\alpha}}\, + (2\pi)^{(1-\alpha)/2} x^{(1-\alpha)/2\alpha} e^{\alpha x^{1/\alpha}},\\ +&\sum \frac{x^{n}}{\Gamma(\alpha n + 1)} \sim (1/\alpha) e^{x^{1/\alpha}},\\ +&\sum e^{-n^{p}}x^{n} +% [** TN: Braces (not parentheses) in sqrt in original] + \sim \sqrt{\frac{2\pi}{p(p - 1)}} \left(\frac{\log x}{p}\right)^{\frac{2-p}{2p-2}} + e^{(p-1)\left(\frac{\log x}{p}\right)^{p/(p-1)}}, +\end{align*} +where $\alpha > 0$ and in the last formula $1 < p < 2$, and throughout $x \to \infty$ +by positive values. + +These results may of course be used to give an upper limit for the +modulus of the particular function considered when $x$~is not necessarily +real, and so for~$M(r)$. Thus in the first case +\[ +M(r) \cleq r^{1/2\alpha} e^{(\alpha/e) x^{1/\alpha}}. +\] + +[The reader who wishes to become familiar with the theory of +integral functions should begin by reading Borel's \textit{Leçons sur les +fonctions entières}. Some additions will be found in the notes at the +end of the same writer's \textit{Leçons sur les fonctions méromorphes}. He +should then read two memoirs by E.~Lindelöf; a short one in the +\textit{Bulletin des Sciences Mathématiques}, t.~27, p.~1, and a long one in +the \textit{Acta Societatis Fennicae}, t.~31, p.~1. Some of the results of this +last paper were proved independently by Boutroux (\textit{Acta Mathematica}, +t.~28, pp.~97 \textit{et~seq.}); but M.~Boutroux's important memoir is largely +occupied by a discussion of some of the most difficult points in the +theory. + +Much of the theory has been developed in a very simple and +elementary way by Pringsheim (\textit{Math.\ Annalen}, Bd.~58, S.~257); and +the reader should certainly consult a short note by Le~Roy (\textit{Bulletin +des Sciences Mathématiques}, t.~24, p.~245). But, after reading the +works of Borel and Lindelöf mentioned above, he will be wise to turn +to Vivanti's \textit{Teoria delle funzioni analitiche} (German translation by +Gutzmer), which contains by far the fullest treatment of the subject +yet published, and an exhaustive list of original memoirs.] +%% -----File: 064.png---Folio 56------- + +\Section[E.]{Power series with a finite radius of convergence.} + +Suppose that $a_{1} + a_{2} + \dots$ is a divergent series: for simplicity +suppose that $a_{n}$~is always positive and steadily increases or decreases +as $n$~increases. Further suppose $e^{-\delta n} \clt a_{n} \clt e^{\delta n}$, so that $\sum a_{n}x^{n}$ is +convergent if $0 \leqq x < 1$. Then a large number of interesting results +have been established connecting the increase of~$a_{n}$, as $n \to \infty$, and +that of $f(x) = \sum a_{n}x^{n}$ as $x \to 1$. The fundamental result is: \emph{if $a_{n} \sim Cb_{n}$, +or, more generally, if $(a_{1} + a_{2} + \dots + a_{n}) \sim C(b_{1} + b_{2} + \dots + b_{n})$, and +$f(x) = \sum a_{n}x^{n}$, $g(x) = \sum b_{n}x^{n}$, then} +\[ +f(x) \sim Cg(x). +\] + +From this theorem it may be deduced that +\begin{align*} +\sum \frac{x^{n}}{n^{p}} + &\sim \frac{\Gamma(1 - p)}{(1 - x)^{1-p}}\quad (p < 1), \\ +F(\alpha, \beta, \gamma, x) + &\sim \frac{\Gamma(\gamma)\, \Gamma(\alpha + \beta - \gamma)}{\Gamma(\alpha)\, \Gamma(\beta)}\, + \frac{1}{(1 - x)^{\alpha+\beta-\gamma}}\quad (\alpha + \beta > \gamma) \\ +F(\alpha, \beta, \alpha + \beta, x) + &\sim \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\, \Gamma(\beta)}\, + l\left(\frac{1}{1 - x}\right). +\end{align*} + +Of further results the following is typical: if +\[ +a_{n} \sim n^{p}/n\, ln \dots l_{m-1}n (l_{m}n)^{q} \dots (l_{m+k}n)^{q_{k}}, +\] +then +\begin{multline*} +F(x) \sim \Gamma(p) \bigg/ \bigg\{(1 - x)^{p+1} \\ + × \frac{1}{1 - x}\, l\frac{1}{1 - x} \dots l_{m-1} \frac{1}{1 - x} + \left(l_{m} \frac{1}{1 - x}\right)^{q}\!\! \dots + \left(l_{m+k} \frac{1}{1 - x}\right)^{q_{k}}\bigg\} +\end{multline*} +if $p > 0$, $q \neq 1$: but +\[ +F(x) \sim 1 \bigg/ + \biggl\{(1 - q)\left(l_{m} \frac{1}{1 - x}\right)^{q-1}\!\! + \biggl(l_{m+1} \frac{1}{1 - x}\biggr)^{q_{1}}\!\! \dots + \biggl(l_{m+k} \frac{1}{1 - x}\biggr)^{q_{k}}\biggl\} +\] +if $p = 0$, $q < 1$ (if $p < 0$ or $p = 0$, $q > 1$, then $\sum a_{n}$ is convergent). + +Thus, \eg +\[ +\sum \frac{n^{p} x^{n}}{(lx)^{q}} + \sim \Gamma(p + 1) \bigg/ + \left\{(1 - x)^{p+1} \left(l \frac{1}{1 - x}\right)^{q}\right\}. +\] +%% -----File: 065.png---Folio 57------- + +As specimens of further results of this character we may quote +\begin{gather*} +x + x^{4} + x^{9} + \dots \sim \tfrac{1}{2} \sqrt{\frac{\pi}{1 - x}}, \\ +x + x^{\alpha} + x^{\alpha^{2}} + \dots + \sim \frac{1}{la}\, l\left(\frac{1}{1 - x}\right)\quad (a > 1), \\ +\sum a^{n} x^{n^{2}} + \sim e\left\{\tfrac{1}{4}\, \frac{(la)^{2}}{l(1/x)}\right\} \quad (a > 1), \\ +\sum e^{n/ln} x^{n} = e_{2}\{u/(1 - x)\} \quad (u \sim 1). +\end{gather*} +Many similar results have been established about series other than +power series: thus +\begin{align*} +\sum \frac{x^{n}}{n(1 + x^{n})} + &\sim \tfrac{1}{2}\, l\left(\frac{1}{1 - x}\right), \\ +\sum \frac{x^{n}}{1 - x^{n}} + &\sim \frac{1}{1 - x}\, l\left(\frac{1}{1 - x}\right). +\end{align*} +As an example of a more precise result we may quote the formula +\[ +\sum \frac{x^{n}}{1 + x^{2n}} + = \tfrac{1}{4} \left\{\frac{\pi}{l(1/x)} - 1\right\} + O\{(1 - x)^{\Delta}\}. +\] +[See + +Bromwich, \textit{Infinite series}, pp.~131 \textit{et~seq.}, 171~\textit{et~seq.}; + +Le~Roy, \textit{Bulletin des Sciences Mathématiques}, t.~24, pp.~245 \textit{et~seq.}; + +Lasker, \textit{Phil.\ Trans.\ Roy.\ Soc.},~(A), vol.~196, p.~433; + +Pringsheim, \textit{Acta Mathematica}, t.~28, p.~1; + +Barnes, \textit{Proc.\ Lond.\ Math.\ Soc.}, vol.~4, p.~284; \textit{Quarterly Journal}, +vol.~37, p.~289; + +Hardy, \textit{Proc.\ Lond.\ Math.\ Soc.}, vol.~3, p.~381; \textit{ibid.},~vol.~5, p.~197; +\textit{ibid.},~vol.~5, p.~342; \\ +where further references will be found. These writers also consider +the extensions of such results to the field of the complex variable.] +%% -----File: 066.png---Folio 58------- + +\Appendix{III.}{Some Numerical Illustrations.} + +Mr J.~Jackson, scholar of Trinity College, has been kind enough to +calculate for me the following numerical results, which will, I think, +be found instructive as comments on some of the matters dealt with in +the body of this tract and in Appendix~II\@. It will of course be understood +that, except in one or two instances, they are approximations +and sometimes quite rough approximations. + +\Section[1.]{Table of the functions $\log x$, $\log\log x$, $\log\log\log x$, etc.} +{ +\[ +\begin{array}{|l|r|r|r|r|r|} +\hline +\CCEntry{x} & \CEntry{\log x} & \CEntry{\log_{2} x} & \CEntry{\log_{3} x} & \CEntry{\log_{4} x} & \CEntry{\log_{5} x}\\ +\hline +\Strut +10 & 2.30 & 0.834 & -0.182 & \Dash & \Dash\\ +10^{3} & 6.91 & 1.933 & 0.659 & -0.417 & \Dash\\ +10^{6} & 13.82 & 2.626 & 0.966 & -0.035 & \Dash\\ +10^{10} & 23.03 & 3.137 & 1.143 & 0.134 & -2.011\\ +10^{15} & 34.54 & 3.542 & 1.265 & 0.235 & -1.449\\ +10^{20} & 46.05 & 3.830 & 1.343 & 0.295 & -1.221\\ +10^{30} & 69.08 & 4.235 & 1.443 & 0.367 & -1.003\\ +10^{60} & 138.15 & 4.928 & 1.595 & 0.467 & -0.762\\ +10^{100} & 230.26 & 5.439 & 1.693 & 0.527 & -0.641\\ +10^{1000} & 2302.58 & 7.742 & 2.047 & 0.716 & -0.334\\ +10^{10^{6}} & 2303 × 10^{3} & 14.650 & 2.685 & 0.987 & -0.013\\ +10^{10^{10}} & 2303 × 10^{7} & 23.860 & 3.172 & 1.154 & 0.144\\ +\hline +\end{array} +\] +} +%% -----File: 067.png---Folio 59------- + +\Section[2.]{Table of the functions $e^{x}$, $e^{e^{x}}$, $e^{e^{e^{x}}}$, etc.} +\[ +\begin{array}{|c|c|c|c|c|c|} +\hline +\CCEntry{x} & \CEntry{ex} & \CEntry{e_{2}x} & \CEntry{e_{3}x} & \CEntry{e_{4}x} \\ +\hline +\Strut +1 & 2.718 & 15.154 & 3,814,260 & 10^{1,656,510}\\ +2 & 7.389 & 1618.2 & 5.85 × 10^{702} & \Dash\\ +3 & 20.085 & 5.28 × 10^{8} & 10^{2.295 × 10^{8}} & \Dash\\ +5 & 148.413 & 2.85 × 10^{64} & 10^{1.24 × 10^{64}} & \Dash\\ +10 & 22026 & 9.44 × 10^{9565} & \Dash & \Dash\\ +\hline +\end{array} +\] + +The function $\log x$ is defined only for $x > 0$, $\log_{2}x$~for $x > 1$, +$\log_{3}x$~for $x > e$, $\log_{4} x$~for $x > e^{e} = e_{2}$, and so on. The values of the +first few numbers $e$,~$e_{2}$, $e_{3}$,~\dots\ are given above, viz.\ $e = 2.718$, $e_{2} = 15.154$, +$e_{3} = 3,814,260$, $e_{4} = 10^{1,656,510}$. + +\Section[3.]{Table of the functions $n!$, $n^{n}$, $n^{n^{n}}$.} + +\[ +\begin{array}{|c|c|c|c|} +\hline +\CCEntry{n} & \CEntry{n!} & \CEntry{n^{n}} & \CEntry{n^{n^{n}}} \\ +\hline +\Strut +1 & 1 & 1 & 1\\ +2 & 2 & 4 & 16\\ +3 & 6 & 27 & 7.634 × 10^{12}\\ +4 & 24 & 256 & 1.491 × 10^{154}\\ +5 & 120 & 3,125 & 9.55 × 10^{2,184}\\ +6 & 720 & 46,656 & 2.7 × 10^{36,305}\\ +7 & 5,040 & 823,543 & 1.4 × 10^{695,974}\\ +8 & 40,320 & \DPtypo{16,827,216}{16,777,216} & 10^{15,151,345}\\ +9 & 362,880 & 3.8742 × 10^{8} & 10^{369,693,100}\\ +10 & 3,628,800 & 10^{10} & 10^{10,000,000,000}\\ +100 & 9.346 × 10^{157} & 10^{200} & \Dash\\ +10^{10} & 10^{9.57 × 10^{10}} & 10^{10^{11}} & \Dash\\ +\hline +\end{array} +\] +%% -----File: 068.png---Folio 60------- + +\Section[4.]{Table to illustrate the convergence of the series\DPtypo{}{.}} + +{\small +\begin{gather*} +\begin{aligned} +&(1)\ \sum_{3}^{\infty} \frac{1}{n\log n (\log\log n)^{2}}. +&&(2)\ \sum_{2}^{\infty} \frac{1}{n(\log n)^{2}}. +&&(3)\ \sum_{1}^{\infty} \frac{1}{n^{s}}\ (s = 1.1).\\ +&(4)\ \sum_{1}^{\infty} \frac{1}{n^{s}}\ (s = 1.5). +&&(5)\ \sum_{1}^{\infty} \frac{1}{n^{s}}\ (s = 2). +&&(6)\ \sum_{1}^{\infty} \frac{1}{n^{s}}\ (s = 10).\\ +&(7)\ \sum_{1}^{\infty} \frac{1}{n^{s}}\ (s = 100). +&&(8)\ \sum_{0}^{\infty} x^{n}\ (x = .9). +&&(9)\ \sum_{0}^{\infty} x^{n}\ (x = .5). +\end{aligned} \\ +\begin{aligned} +&(10)\ \sum_{0}^{\infty} x^{n}\ (x = .1). +&&(11)\ 1 + \frac{1}{2!} + \frac{1}{3!} + \dots. +&&(12)\ 1 + \frac{1}{2^{2}} + \frac{1}{3^{3}} + \dots.\\ +&(13)\ \sum_{0}^{\infty} x^{n^{2}}\ (x = .9). +&&(14)\ \sum_{0}^{\infty} x^{n^{2}}\ (x = .5) +&&(15)\ \sum_{0}^{\infty} x^{n^{2}}\ (x = .1). +\end{aligned} \\ +(16)\ \frac{1}{1^{1^{1}}} + \frac{1}{2^{2^{2}}} + \frac{1}{3^{3^{3}}} +\dots. +\end{gather*} +\footnotesize\settowidth{\TmpLen}{calculate the sum correctly to}% +\[ +%[** TN: Force centering of slightly over-wise table] +\makebox[0pt][c]{$ +\begin{array}{|c|c|c|c|c|c|} +\hline +&&\multicolumn{4}{|c|}{% + \parbox{\TmpLen}{% + \centering\footnotesize\medskip Number of terms required to\\ + calculate the sum correctly to}} \\ +\text{Series} & \text{Sum} & 2 & 10 & 100 & 1000 \\ +&&&\multicolumn{2}{|c|}{\centering\text{\footnotesize decimal places\footnotemark}} & \\ +\hline +\Strut +1 & 38.43 & 10^{3.14 × 10^{86}} & \Dash & \Dash & \Dash\\ +2 & 2.11 & 7.23 × 10^{86} & 10^{8.6 × 10^{9}} & \Dash & \Dash\\ +3 & 10.58 & 10^{33} & 10^{113} & 10^{1013} & 10^{10013}\\ +4 & 2.612 & 160,000 & 16 × 10^{20} & 6 ×10^{200} & 16 ×10^{2000}\\ +5 & \frac{1}{6}\pi^{2} = 1.64493 & 200 & 2 × 10^{10} & 2 × 10^{100} & 2 × 10^{1000}\\ +6 & 1.0009846 & 1 & 11 & 1.093 × 10^{11} & 1.093 × 10^{111}\\ +7 & 1 + (1.27 × 10^{-30}) & 1 & 1 & 10 & 1.213 × 10^{10}\\ +8 & 10 & 73 & 247 & 2214 & 21883\\ +9 & 2 & 9 & 36 & 336 & 3325\\ +10 & 10/9 & 3 & 11 & 101 & 1001\\ +11 & e - 1 = 1.718282 & 5 & 13 & 70 & 440\\ +12 & 1.291286 & 3 & 10 & 57 & 386\\ +13 & 3.234989 & 8 & 15 & 46 & 148\\ +14 & 1.564468 & 3 & 6 & 19 & 58\\ +15 & 1.100100 & 2 & 4 & 11 & 32\\ +16 & 1.062500 & 2 & 2 & 3 & 4\\ +\hline +\end{array}$} +\]} +\footnotetext{The phrase `calculate the sum correctly to $m$~decimal places' is used as + equivalent to `calculate with an error less than $\frac{1}{2} × 10^{-m}$.' In the case of a very + slowly convergent series the interpretation affects the numbers to a considerable + extent. The numbers would be considerably more difficult to calculate were the + phrase interpreted in its literal sense.}% +%% -----File: 069.png---Folio 61------- + +Such a series as~(7) is of course exceedingly rapidly convergent \emph{at +first}, \ie\ a very few terms suffice to give the sum correctly to a considerable +number of places; but if the sums are wanted to a very large +number of places, even the series~(8) proves to be far more practicable. + +Mr William Shanks (\textit{Proc.\ Roy.\ Soc.}, vol.~21, p.~318) calculated +the value of~$\pi$ to $707$~places of decimals from Machin's formula +\[ +\pi = 16\left(\frac{1}{5} - \frac{1}{3·5^{3}} + \frac{1}{5·5^{5}} - \dots\right) + - 4 \left(\frac{1}{239} - \frac{1}{3·239^{3}} + \dots\right). +\] +He does not state the number of terms he found it necessary to use, +but, in a previous calculation to $530$~places, used $747$~terms of the +first and $219$~terms of the second series. He also (\textit{ibid.}, vol.~6, p.~397) +calculated~$e$ to $205$~places from the series~(11). + + +\Section[5.]{Table to illustrate the divergence of the series} + +{\small +\begin{align*} +&(1)\ \frac{1}{\log \log 3} + \frac{1}{\log \log 4} + \dots. +&&(2)\ \frac{1}{\log 2} + \frac{1}{\log 3} + \dots.\\ +&(3)\ 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \dots. +&&(4)\ 1 + \frac{1}{2} + \frac{1}{3} + \dots.\\ +&(5)\ \frac{1}{2\log 2} + \frac{1}{3\log 3} + \dots. +&&(6)\ \frac{1}{3\log 3\log\log 3} + \frac{1}{4\log 4 \log\log 4} \DPtypo{}{{}+ \dots}. +\end{align*} +\[ +\begin{array}{|c|c|c|c|c|c|c|} +\hline +&\multicolumn{6}{|c|}{% + \text{\footnotesize Number of terms required to make the sum greater than}} \\ +\text{Series} & 3 & 5 & 10 & 100 & 1000 & 10^{6}\\ +\hline +\Strut +1 & 1 & 1 & 1 & 116 & 1800 & 2.6 × 10^{6}\\ +2 & 3 & 7 & 20 & 440 & 7600 & 1.5 × 10^{7}\\ +3 & 5 & 10 & 33 & 2500 & 2.5 × 10^{5} & 2.5 × 10^{11}\\ +4 & 11 & 82 & 12390 & 10^{43} & 10^{.43 ×10^{3}} & 10^{.43 × 10^{6}}\\ +5 & 8690 & 1.3 × 10^{29} & 10^{4300} & 10^{5 × 10^{42}} & \Dash & \Dash\\ +6 & 1 & 60 \text{ \emph{to} } 70 & 10^{10^{100}} & \Dash & \Dash & \Dash\\ +\hline +\end{array} +\]} + +\Section[6.]{Roots of certain equations.} + +\Item{(i)} The equation $e^{x} = x^{1,000,000}$ has a root just larger than unity (the +excess of the root over unity being practically~$10^{-6}$) and a large root +in the neighbourhood of~$16,610,800$. The equation $e^{x} = 1,000,000 x^{1,000,000}$ +has roots nearly equal to those of the above. The one near unity is +practically $12.82 x 10^{-6}$ less than unity, while the large root exceeds +the root of the above equation by about~$13.82$. +%% -----File: 070.png---Folio 62------- + +\Item{(ii)} The equation $e^{x^{2}} = x^{10^{10}}$ has a root somewhere near~$357,500$. + +\Item{(iii)} {\Loosen The equation $e^{e^{x}} = 10^{10} x^{10} e^{10^{10} x^{10}}$ has a root near~$64.7$. The +root differs by less than~$10^{-26}$ from the corresponding root of $e^{x} = 10^{10} x^{10}$. +The corresponding root of $e^{x} = x^{10}$ is about~$35.8$.} + +\Item{(iv)} The positive roots of $x^{x^{x}} = 1,000,000$ and of $x^{x^{x}} = 10^{1,000,000}$ are +approximately $2.68$~and~$7.11$. + +\Item{(v)} If $x^{10} = 10^{y}$, then for $x = 100$, $y = 20$; and for $x = 10^{10}$, $y = 100$. + +If $x^{10^{10}} = 10^{10^{y}}$, then for $x = 100$, $y = 10.30$; for $x = 10^{10}$, $y = 11$; and +for $x = 10^{10^{10}}$, $y = 20$. + +If $x^{10^{10^{10}}} = 10^{10^{10^{y}}}$, then for $x = 10^{10}$, $y = 10 + (4.3 × 10^{-11})$; for $x = 10^{10^{10}}$, +$y = 10 + (4.3 × 10^{-10})$; and for $x = 10^{10^{10^{10}}}$, $y = 10.30$. + +\Section[7.]{Some numbers of physics.} + +The distance to $\alpha$~Centauri is roughly $26,000,000,000,000$ miles or +$1.65 × 10^{18}$~inches. The number of inches lies between $19!$~and~$20!$ and +is approximately equal to~$e^{e^{3.74}}$ or~$16^{e^{e}}$. Again, writing $15$~letters to the +inch (an average size in print) a line to the star would be sufficient +for the writing at length of~$10^{2.47 × 10^{19}}$. The latter number is approximately +equal to $(14 × 10^{17})!$, $e^{e^{e^{3.83}}}$, or $(10^{6.5 × 10^{12}})^{e^{e^{e}}}$. + +If we take the distance to the end of the visible universe to be that +through which light travels in $10,000$~years, we find that this distance +when expressed in wave-lengths of sodium light is measured roughly +by the numbers +\[ +1.6 ×10^{26},\quad 26!,\quad e^{e^{4.10}},\quad (53.6)^{e^{e}},\quad 3.29^{3.29^{3.29}}. +\] + +If we assume the average distance between the centres of two +adjacent molecules of the earth's substance to be $10^{-8}$~cm., we find +that the number of molecules in the earth is roughly +\[ +10.8 × 10^{50},\quad 42!,\quad e^{e^{4.77}},\quad (2333)^{e^{e}},\quad 3.56^{3.56^{3.56}}. +\] +\vfill +\hrule +\Strut[8pt] +{\scriptsize CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS} +%% -----File: 071.png---Folio 63------- +%% -----File: 072.png---Folio 64------- +%% -----File: 073.png---Folio 65------- +%% -----File: 074.png---Folio 66------- +\clearpage +\thispagestyle{empty} +\begin{center} +\Titlefont{Cambridge Tracts in Mathematics\\ + and Mathematical Physics}\\ +\rule{1.5in}{1.0pt} +\end{center} + +{\footnotesize +\Catalog{No.~1.} VOLUME AND SURFACE INTEGRALS USED IN +PHYSICS, by \textsc{J.~G. 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Grace}, M.A., +F.R.S. + +\Inprep AN INTRODUCTION TO THE THEORY OF ATTRACTIONS, +by \textsc{T.~J.~I'A. Bromwich}, M.A., F.R.S.\par +} +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% +\FlushRunningHeads +\vfill +\PGLicense +\begin{PGtext} +End of Project Gutenberg's Orders of Infinity, by Godfrey Harold Hardy + +*** END OF THIS PROJECT GUTENBERG EBOOK ORDERS OF INFINITY *** + +***** This file should be named 38079-pdf.pdf or 38079-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/8/0/7/38079/ + +Produced by Andrew D. 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Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..4d30144 --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #38079 (https://www.gutenberg.org/ebooks/38079) |
