\chapter[A sketch of pure complex theory]%
  {A bare sketch of the pure theory of the complex variable}
\label{purec}
\index{variable!complex}
\index{complex variable}
\index{mathematics!professional or pure}
\index{professional mathematics}
\index{pure mathematics}

At least three of the various disciplines of pure mathematics stand out
for their pedagogical intricacy and the theoretical depth of their core
results.  The first of the three is number theory which, except for the
simple results of \S~\ref{noth:220}, scientists and engineers tend to
get by largely without.  The second is matrix theory (chapters~\ref{matrix}
through~\ref{eigen}), a bruiser of a discipline the applied
mathematician of the computer age---try though he might---can hardly
escape.  The third is the pure theory of the complex variable.

The introduction's \S~\ref{intro:310} admires the beauty of the pure
theory of the complex variable even while admitting that that
theory's
``arc regrettably takes off too late and flies too far from
applications for such a book as this.''
To develop the pure theory properly is a worthy book-length
endeavor of its own requiring moderately advanced preparation on its
reader's part which, however, the reader who has reached the end of the
present book's chapter~\ref{inttx} possesses.  If the writer doubts the
strictly applied \emph{necessity} of the pure theory, still, he does not
doubt its health to one's overall mathematical formation.  It provides
another way to think about complex numbers.  Scientists and engineers
with advanced credentials occasionally expect one to be acquainted with
it for technical-social reasons, regardless of its practical use.
Besides, the pure theory is interesting.  This alone recommends some
attention to it.

The pivotal result of pure complex-variable theory is the Taylor series
by Cauchy's impressed residue theorem.  If we will let these few pages
of appendix replace an entire book on the pure theory, then Cauchy's and
Taylor's are the results we will sketch.  The bibliography lists
presentations far more complete.
%(This presentation, as advertised, is just a sketch.)
%(The reader who has reached the end of the
%(chapter~\ref{inttx} will understand already why the presentation is strictly
%optional, interesting maybe but deemed unnecessary to the book's applied
%mathematical development.)

\index{Cauchy's impressed residue theorem}
\index{impressed residue theorem, Cauchy's}
\index{residue theorem, Cauchy's impressed}
\index{cleverness}
%\index{Arnold, D.~N.}
\emph{Cauchy's impressed residue theorem}%
\footnote{
  This is not a standard name.  Though they name various associated
  results after Cauchy in one way or another, neither~\cite{Hildebrand}
  nor~\cite{Arnold:1997} seems to name this particular result, though
  both do feature it.  Since~(\ref{purec:100:10}) impresses a pole and
  thus also a residue on a function $f(z)$ which in the domain of
  interest lacks them, the name \emph{Cauchy's impressed residue
  theorem} ought to serve this appendix's purpose ably enough.
}
is that
\bq{purec:100:10}
  f(z) = \frac{1}{i2\pi} \oint \frac{f(w)}{w-z} \,dw
\eq
if~$z$ lies within the closed complex contour about which the integral
is taken and if $f(z)$ is everywhere analytic (\S~\ref{taylor:320})
within and along the contour.  More than one proof of the theorem is
known, depending on the assumptions from which the mathematician prefers
to start, but this writer is partial to an instructively clever proof he
has learned from D.~N.\ Arnold%
\footnote{\cite[\S~III]{Arnold:1997}}
which goes as follows.  Consider the function
\[
  g(z,t) \equiv \frac{1}{i2\pi} \oint \frac{f[z+(t)(w-z)]}{w-z} \,dw,
\]
whose derivative with respect to the parameter~$t$ is%
\footnote{
  The book does not often employ Newton's notation $f'(\cdot) \equiv
  [(d/d\zeta)f(\zeta)]_{\zeta=(\cdot)}$ of
  \S~\ref{drvtv:240} but the notation is handy here because it evades
  the awkward circumlocution of changing $\zeta \la z$
  in~(\ref{purec:100:10}) and then writing,
  \[
    \frac{\pl g}{\pl t} = \frac{1}{i2\pi} \oint \frac{[(d/d\zeta)f(\zeta)]_{\zeta=z+(t)(w-z)}}{w-z} \,dw.
  \]
}
\[
  \frac{\pl g}{\pl t}
    = \frac{1}{i2\pi} \oint f'[z+(t)(w-z)] \,dw.
\]
We notice that this is
\bqb
  \frac{\pl g}{\pl t}
  &=&
    \frac{1}{i2\pi} \oint \frac{\pl}{\pl w}\left\{
      \frac{f[z+(t)(w-z)]}{t}
    \right\} \,dw
  \\&=&
    \frac{1}{i2\pi} \left\{
      \frac{f[z+(t)(w-z)]}{t}
    \right\}_{w=a}^b,
\eqb
where~$a$ and~$b$ respectively represent the contour integration's
beginning and ending points.  But this integration ends where it begins
and its integrand (lacking a~$w$ in the denominator) is analytic within
and along the contour, so $a=b$ and the factor~$\{\cdot\}_{w=a}^b$ in
braces vanishes, whereupon
\[
  \frac{\pl g}{\pl t} = 0,
\]
meaning that $g(z,t)$ does not vary with~$t$.  Observing
per~(\ref{taylor:cauchy0}) that
\[
  \frac{1}{i2\pi} \oint \frac{dw}{w-z} = 1,
\]
we have that
\[
  f(z)
  = \frac{f(z)}{i2\pi} \oint \frac{dw}{w-z}
  = g(z,0) = g(z,1)
  =
  \frac{1}{i2\pi} \oint \frac{f(w)}{w-z} \,dw
\]
as was to be proved.  (There remains a basic question as to whether the
paragraph's integration is even valid.  Logically, it ought to be valid,
since $f[z]$ being analytic is infinitely differentiable,%
\footnote{
  The professionals minimalistically actually require only that the
  function be once differentiable under certain conditions, from which
  they prove infinite differentiability, but this is a fine point which
  will not concern us here.
}
but when the integration is used as the sole theoretical support for the
entire calculus of the complex variable, well, it seems an awfully
slender reed to carry so heavy a load.  Admittedly, maybe this is only a
psychological problem, but a professional mathematician will devote many
pages to preparatory theoretical constructs before even attempting the
integral, the result of which lofty effort is not in the earthier spirit
of applied mathematics.  On the other hand, now that the reader has
followed the book along its low road and the high integration is given
only in reserve, now that the integration reaches a conclusion already
believed and, once there, is asked to carry the far lighter load of this
appendix only, the applied reader may feel easier about trusting it.)

\index{Goursat, Edouard (1858--1936)}
%\index{Hildebrand, F.~B.}
One could follow Arnold hence toward the proof of the theorem of one
Goursat and further toward various other interesting results, a path of
study the writer recommends to sufficiently interested readers:
see~\cite{Arnold:1997}.  Being in a tremendous hurry ourselves, however,
we will leave Arnold and follow F.~B.\ Hildebrand%
\footnote{\cite[\S~10.7]{Hildebrand}}
directly toward the Taylor series.  Positing some expansion point~$z_o$
and then expanding~(\ref{purec:100:10}) geometrically
per~(\ref{alggeo:228:45}) about it, we have that
\bqb
  f(z)
  &=&
    \frac{1}{i2\pi} \oint \frac{f(w)}{(w-z_o)-(z-z_o)} \,dw
  \\&=&
    \frac{1}{i2\pi} \oint \frac{f(w)}{(w-z_o)[1 - (z-z_o)/(w-z_o)]} \,dw
  \\&=&
    \frac{1}{i2\pi} \oint \frac{f(w)}{w-z_o}
    \sum_{k=0}^\infty \left[\frac{z-z_o}{w-z_o}\right]^k \,dw
  \\&=&
    \sum_{k=0}^\infty \left\{
      \left[
        \frac{1}{i2\pi}
        \oint \frac{f(w)}{(w-z_o)^{k+1}} \,dw
      \right] (z-z_o)^k
    \right\},
\eqb
which, being the power series
\bq{purec:100:40}
  \begin{split}
    f(z) &= \sum_{k=0}^\infty (a_k) (z-z_o)^k, \\
    a_k &\equiv \frac{1}{i2\pi} \oint \frac{f(w)}{(w-z_o)^{k+1}} \,dw,
  \end{split}
\eq
by definition constitutes the Taylor series~(\ref{taylor:310:20}) for
$f(z)$ about $z=z_o$, assuming naturally that $\left|z-z_o\right| <
\left|w-z_o\right|$ for all~$w$ along the contour so that the geometric
expansion above will converge.

The important theoretical implication of~(\ref{purec:100:40}) is that
\emph{every function has a Taylor series about any point across whose
immediate neighborhood the function is analytic.}  There evidently is no
such thing as an analytic function without a Taylor series---a fact we
already knew if we have read and believed chapter~\ref{taylor}, but some
readers may find it more convincing this way.

Comparing~(\ref{purec:100:40}) against~(\ref{taylor:310:20}),
incidentally, we have also that
\bq{purec:100:50}
    \left.\frac{d^k\!f}{dz^k}\right|_{z=z_o}
    = \frac{k!}{i2\pi} \oint \frac{f(w)}{(w-z_o)^{k+1}} \,dw,
\eq
which is an alternate way to write~(\ref{taylor:350:30}).

Close inspection of the reasoning by which we have
reached~(\ref{purec:100:40}) reveals, quite by the way, at least one
additional result which in itself tends to vindicate the pure theory's
technique.  It is this: that \emph{a Taylor series remains everywhere
valid out to the distance of the nearest nonanalytic point.}  The
proposition is explained and proved as follows.  For the aforementioned
contour of integration nothing prevents one from choosing a circle,
centered in the Argand plane on the expansion point $z=z_o$, the
circle's radius just as large as it can be while still excluding all
nonanalytic points.  The requirement that $\left|z-z_o\right| <
\left|w-z_o\right|$ for all~$w$ along the contour evidently is met for
all~$z$ inside such a circle, which means that the Taylor
series~(\ref{purec:100:40}) converges for all~$z$ inside the circle,
which---precisely because we have stipulated that the circle be the
largest possible centered on the expansion point---implies and thus
proves the proposition in question.  As an example of the proposition's
use, consider the Taylor series Table~\ref{taylor:315:tbl} gives for
$-\ln (1-z)$, whose nearest nonanalytic point at $z=1$ lies at unit
distance from the series' expansion point $z=0$: according to the result
of this paragraph, the series in question remains valid over the Argand
circle out to unit distance, $|1-z| < 1$.

% ----------------------------------------------------------------------

