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

\chapter{Introduction to special functions}
\label{specf}
\index{special functions}

% diagn
[This chapter is yet only a stub of a draft.]

No topic more stirs the pure mathematician's imagination than that of
number theory, so briefly addressed by this book's \S~\ref{noth:220}.
So it is said.  However, an applied mathematician is pleased to follow a
lesser muse, and the needful topic that most provokes his mathematical
curiosity may be that of special functions.

What is a \emph{special function?}  Trouble arises at once, before the
first mathematical symbol strikes the page, for it is not easy to
discover a precise definition of the term.  N.~N.\ Lebedev and Larry~C.
Andrews, authors respectively in Russian and English of two of the
better post-World War~II books on the topic,%
\footnote{
  The books respectively are \cite{Lebedev} and~\cite{Andrews}, the
  latter originally in English, the former available at the time of this
  writing in Richard~A. Silverman's excellent, inexpensive English
  translation.
}
seem to decline to say what a special function \emph{is,} merely
suggesting in their respective prefaces some things it \emph{does,}
diving thence fluidly into the mathematics before anyone should notice
that those authors have motivated but never quite delineated their topic
(actually, it's not a bad approach, and reading Lebedev or Andrews one
does soon perceive the shape of the thing).  Abramowitz and Stegun,
whose celebrated handbook%
\footnote{\cite{A-S}}
though not expressly a book on special functions is largely about them,
are even silenter on the question.  The sober dictionary%
\footnote{\cite{Webster}}
on the author's desk correctly defines such mathematical terms as
``eigenvalue'' and ``Fourier transform'' but seems unacquainted with
``special function.''  What are we to make of this?

\index{elementary function}
\index{canonical form}
\index{integral}
\index{integral equation}
\index{differential equation}
Here is what.  A \emph{special function} is an analytic function
(\S~\ref{taylor:320})---likely of a single and at most of a few complex
scalar variables, harder to analyze and evaluate than the
\emph{elementary functions} of chapters~\ref{alggeo} through~\ref{cexp},
defined in a suitably canonical form---that serves to evaluate an
integral, to solve an integral equation,%
\footnote{
  It need not interest you but in case it does, an \emph{integral
  equation} is an equation like
  \[
    \int_{-\infty}^\infty g(z,w) f(w) \,dw = h(z),
  \]
  in which the unknown is not a variable but a function $f(w)$ that
  operates on a dummy variable of integration.  Actually, we have
  already met integral equations in disguise, in discretized form, in
  matrix notation (chapters~\ref{matrix} through~\ref{eigen}) resembling
  \[
    G\ve f = \ve h,
  \]
  which means no more than it seems to mean; so maybe integral equations
  are not so strange as they look.  The sample integral equation is just the
  matrix equation with the discrete vectors~$\ve f$ and~$\ve h$ replaced
  by their continuous versions $\Delta w\,f(j\,\Delta w)$ and $\Delta
  z\,h(i\,\Delta z)$ (the~$i$ representing not the imaginary unit here
  but just an index, as in chapter~\ref{matrix}).
}
or to solve a differential equation elementary functions alone cannot
evaluate or solve.  Such a definition approximates at least the aspect
and use of the special functions this book means to treat.  The
definition will do to go on with.

The fascination of special functions to the scientist and engineer lies
in how gracefully they analyze otherwise intractable physical models; in
how reluctantly they yield their mathematical secrets; in how readily
they conform to unexpected applications; in how often they connect
seemingly unrelated phenomena; and in that, the more intrepidly one
explores their realm, the more disquietly one feels that one had barely
penetrated the realm's frontier.  The topic of special functions seems
inexhaustible.  We surely will not begin to exhaust the topic in this
book; yet, even so, useful results will start to flow from it almost at
once.

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

\section{The Gaussian pulse and its moments}
\label{specf:220}
\index{Gaussian pulse}
\index{pulse, Gaussian}

We have already met the versatile Gaussian pulse,
\[
  \Omega(t) = \frac{\exp\left(-t^2/2\right)}{\sqrt{2\pi}},
\]
in \S~\ref{fouri:130} and, earlier, in Fig.~\ref{fours:095:fig1}.
Chapters~\ref{fours} and~\ref{fouri}, treating the series and transform
of Fourier, have chiefly considered real~$t$, but one can more generally
change $z \la t$ to give the function
\bq{specf:220:gauss}
  \Omega(z) \equiv \frac{\exp\left(-z^2/2\right)}{\sqrt{2\pi}}
\eq
a complex argument~$z$.

[Chapter to be \mbox{continued\ldots.}]

