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% The Project Gutenberg EBook of Orders of Infinity, by Godfrey Harold Hardy
%                                                                         %
% This eBook is for the use of anyone anywhere at no cost and with        %
% almost no restrictions whatsoever.  You may copy it, give it away or    %
% re-use it under the terms of the Project Gutenberg License included     %
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%                                                                         %
% Title: Orders of Infinity                                               %
%        The 'Infinitärcalcül' of Paul Du Bois-Reymond                    %
%                                                                         %
% Author: Godfrey Harold Hardy                                            %
%                                                                         %
% Release Date: November 25, 2011 [EBook #38079]                          %
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% Language: English                                                       %
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% *** START OF THIS PROJECT GUTENBERG EBOOK ORDERS OF INFINITY ***        %
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\begin{PGtext}
The Project Gutenberg EBook of Orders of Infinity, by Godfrey Harold Hardy

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever.  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}
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\end{center}
\clearpage

%%%% Credits and transcriber's note %%%%
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\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)
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\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. Leatham}, M.A. \textit{$2$s.\ $6$d.\ net.}

\Catalog{No.~2.} THE INTEGRATION OF FUNCTIONS OF A SINGLE
VARIABLE, by \textsc{G.~H. Hardy}, M.A., F.R.S. \textit{$2$s.\ $6$d.\ net.}

\Catalog{No.~3.} QUADRATIC FORMS AND THEIR CLASSIFICATION
BY MEANS OF INVARIANT FACTORS, by
\textsc{T.~J.~I'A. Bromwich}, M.A., F.R.S. \textit{$3$s.\ $6$d.\ net.}

\Catalog{No.~4.} THE AXIOMS OF PROJECTIVE GEOMETRY, by
\textsc{A.~N. Whitehead}, Sc.D., F.R.S. \textit{$2$s.\ $6$d.\ net.}

\Catalog{No.~5.} THE AXIOMS OF DESCRIPTIVE GEOMETRY, by
\textsc{A.~N. Whitehead}, Sc.D., F.R.S. \textit{$2$s.\ $6$d.\ net.}

\Catalog{No.~6.} ALGEBRAIC EQUATIONS, by \textsc{G.~B. Mathews}, M.A.,
F.R.S. \textit{$2$s.\ $6$d.\ net.}

\Catalog{No.~7.} THE THEORY OF OPTICAL INSTRUMENTS, by
\textsc{E.~T. Whittaker}, M.A., F.R.S. \textit{$2$s.\ $6$d.\ net.}

\Catalog{No.~8.} THE ELEMENTARY THEORY OF THE SYMMETRICAL
OPTICAL INSTRUMENT, by \textsc{J.~G. Leatham},
M.A. \textit{$2$s.\ $6$d.\ net.}

\Catalog{No.~9.} INVARIANTS OF QUADRATIC DIFFERENTIAL
FORMS, by \textsc{J.~E. Wright}, M.A. \textit{$2$s.\ $6$d.\ net.}

\Catalog{No.~10.} AN INTRODUCTION TO THE STUDY OF INTEGRAL
EQUATIONS, by \textsc{Maxime Bôcher}, B.A., Ph.D. \textit{$2$s.\ $6$d.\
net.}

\Catalog{No.~11.} THE FUNDAMENTAL THEOREMS OF THE DIFFERENTIAL
CALCULUS, by \textsc{W.~H. Young}, Sc.D.,
F.R.S. \textit{$2$s.\ $6$d.\ net.}

\Catalog{No.~12.} ORDERS OF INFINITY: \textsc{The 'Infinitärcalcül' of Paul
du~Bois-Reymond}, by \textsc{G.~H. Hardy}, M.A., F.R.S.
\textit{$2$s.\ $6$d.\ net.}

\subsubsection*{\normalfont\itshape In preparation.}

\Inprep THE DEFINITE INTEGRAL, ITS MEANING AND
FUNDAMENTAL PROPERTIES, by \textsc{E.~W. Hobson},
Sc.D., F.R.S.

\Inprep SINGULAR POINTS AND ASYMPTOTES OF PLANE
CURVES, by Miss \textsc{C.~A. Scott}.

\Inprep THE CIRCLE AT INFINITY, by \textsc{J.~H. 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
}
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 *File List*
    book.cls    2007/10/19 v1.4h Standard LaTeX document class
    bk12.clo    2007/10/19 v1.4h Standard LaTeX file (size option)
inputenc.sty    2008/03/30 v1.1d Input encoding file
  latin1.def    2008/03/30 v1.1d Input encoding file
  ifthen.sty    2001/05/26 v1.1c Standard LaTeX ifthen package (DPC)
 amsmath.sty    2000/07/18 v2.13 AMS math features
 amstext.sty    2000/06/29 v2.01
  amsgen.sty    1999/11/30 v2.0
  amsbsy.sty    1999/11/29 v1.2d
  amsopn.sty    1999/12/14 v2.01 operator names
 amssymb.sty    2009/06/22 v3.00
amsfonts.sty    2009/06/22 v3.00 Basic AMSFonts support
   alltt.sty    1997/06/16 v2.0g defines alltt environment
footmisc.sty    2009/09/15 v5.5a a miscellany of footnote facilities
indentfirst.sty    1995/11/23 v1.03 Indent first paragraph (DPC)
  icomma.sty    2002/03/10 v2.0 (WaS)
    calc.sty    2007/08/22 v4.3 Infix arithmetic (KKT,FJ)
  yfonts.sty    2003/01/08 v1.3 (WaS)
fancyhdr.sty    
graphicx.sty    1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR)
  keyval.sty    1999/03/16 v1.13 key=value parser (DPC)
graphics.sty    2009/02/05 v1.0o Standard LaTeX Graphics (DPC,SPQR)
    trig.sty    1999/03/16 v1.09 sin cos tan (DPC)
graphics.cfg    2009/08/28 v1.8 graphics configuration of TeX Live
  pdftex.def    2009/08/25 v0.04m Graphics/color for pdfTeX
 caption.sty    2009/10/09 v3.1k Customizing captions (AR)
caption3.sty    2009/10/09 v3.1k caption3 kernel (AR)
geometry.sty    2008/12/21 v4.2 Page Geometry
   ifpdf.sty    2009/04/10 v2.0 Provides the ifpdf switch (HO)
  ifvtex.sty    2008/11/04 v1.4 Switches for detecting VTeX and its modes (HO)
geometry.cfg
hyperref.sty    2009/10/09 v6.79a Hypertext links for LaTeX
 ifxetex.sty    2009/01/23 v0.5 Provides ifxetex conditional
 hycolor.sty    2009/10/02 v1.5 Code for color options of hyperref/bookmark (HO
)
xcolor-patch.sty    2009/10/02 xcolor patch
  pd1enc.def    2009/10/09 v6.79a Hyperref: PDFDocEncoding definition (HO)
etexcmds.sty    2007/12/12 v1.2 Prefix for e-TeX command names (HO)
infwarerr.sty    2007/09/09 v1.2 Providing info/warning/message (HO)
hyperref.cfg    2002/06/06 v1.2 hyperref configuration of TeXLive
kvoptions.sty    2009/08/13 v3.4 Keyval support for LaTeX options (HO)
kvsetkeys.sty    2009/07/30 v1.5 Key value parser with default handler support 
(HO)
     url.sty    2006/04/12  ver 3.3  Verb mode for urls, etc.
  bitset.sty    2007/09/28 v1.0 Data type bit set (HO)
 intcalc.sty    2007/09/27 v1.1 Expandable integer calculations (HO)
bigintcalc.sty    2007/11/11 v1.1 Expandable big integer calculations (HO)
pdftexcmds.sty    2009/09/23 v0.6 LuaTeX support for pdfTeX utility functions (
HO)
ifluatex.sty    2009/04/17 v1.2 Provides the ifluatex switch (HO)
 ltxcmds.sty    2009/08/05 v1.0 Some LaTeX kernel commands for general use (HO)

atbegshi.sty    2008/07/31 v1.9 At begin shipout hook (HO)
 hpdftex.def    2009/10/09 v6.79a Hyperref driver for pdfTeX
    umsa.fd    2009/06/22 v3.00 AMS symbols A
    umsb.fd    2009/06/22 v3.00 AMS symbols B
supp-pdf.mkii
   color.sty    2005/11/14 v1.0j Standard LaTeX Color (DPC)
   color.cfg    2007/01/18 v1.5 color configuration of teTeX/TeXLive
 nameref.sty    2007/05/29 v2.31 Cross-referencing by name of section
refcount.sty    2008/08/11 v3.1 Data extraction from references (HO)
 38079-t.out
 38079-t.out
./images/cups.png
  omscmr.fd    1999/05/25 v2.5h Standard LaTeX font definitions
./images/fig1.pdf
./images/fig2.pdf
./images/fig3.pdf
./images/fig4.pdf
./images/fig5.pdf
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