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\author{Robin K. S. Hankin}
\title{Cauchy's integral theorem: a numerical perspective}
%\VignetteIndexEntry{The residue theorem from a numerical perspective}
%% for pretty printing and a nice hypersummary also set:
%% \Plainauthor{Achim Zeileis, Second Author} %% comma-separated
\Plaintitle{The residue theorem from a numerical perspective}


%% for pretty printing and a nice hypersummary also set:
\Plainauthor{Robin K. S. Hankin}

%% an abstract and keywords

\Abstract{Here I use the {\tt myintegrate()} function of the
  \pkg{elliptic} package to illustrate three classical theorems from
  analysis: Cauchy's integral theorem, the residue theorem, and
  Cauchy's integral formula.  In all cases, numerical values agree
  with analytical results to within numerical precision.  Some further
  work is suggested.
}

\Keywords{Residue theorem, Cauchy formula, Cauchy's integral formula,
  contour integration, complex integration, Cauchy's theorem}


\Keywords{Elliptic functions, residue theorem, numerical integration, \proglang{R}}
\Plainkeywords{Elliptic functions, residue theorem, numerical integration, R}

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\Address{
  Robin K. S. Hankin\\
  University of Stirling\\
  Scotland\\
  email: \email{hankin.robin@gmail.com}
}


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\begin{document}

<<requirepackage,echo=FALSE,print=FALSE>>=
require(elliptic,quietly=TRUE)
@ 

\setlength{\intextsep}{0pt}
\begin{wrapfigure}{r}{0.2\textwidth}
\begin{center}
\includegraphics[width=1in]{\Sexpr{system.file("help/figures/elliptic.png",package="elliptic")}}
\end{center}
\end{wrapfigure}

\section{Introduction}

Cauchy's integral theorem and its corollaries are some of the most
startling and fruitful ideas in the whole of mathematics.  They place
powerful constraints on analytical functions; and show that a
function's local behaviour dictates its global properties.
Cauchy's integral theorem may be used to prove the residue theorem and
Cauchy's integral formula; these three theorems form a powerful and
cohesive suite of results.

In this short document I use numerical methods to illustrate and
highlight some of their consequences for complex analysis.

\subsection{Cauchy's integral theorem}

Augustin-Louis Cauchy proved an early version of the integral theorem
in 1814; it required that the function's derivative was continuous.
This assumption was removed in 1900 by \'Edouard Goursat at the
expense of a more difficult proof; the result is sometimes known as
the Cauchy-Goursat theorem and is now a cornerstone of complex analysis.
Formally, in modern notation, we have:

\noindent {\bf Cauchy's integral theorem}.  If $f(z)$ is holomorphic
in a simply connected domain $\Omega\subset\mathbb{C}$, then for any
closed contour $C$ in $\Omega$,

$$\int_{C}f(z)\,dz=0.$$
\\

To demonstrate this theorem numerically, I will use the integration
suite of functions provided with the \pkg{elliptic} package which
perform complex integration of a function along a path specified
either as a sequence of segments [{\tt integrate.segments()}] or a
curve [{\tt integrate.contour()}].  

Let us consider $f(z)=\exp z$, holomorphic over all of $\mathbb{C}$,
and evaluate

$$\oint_C f(z)\,dz$$

where $C$ is the square $0\longrightarrow 1\longrightarrow
1+i\longrightarrow i\longrightarrow 0$ (figure~\ref{square}).  Numerically:

\begin{figure}\centering
\begin{tikzpicture}
% Axes
\draw [help lines,->] (-1,  0) -- (2, 0);
\draw [help lines,->] ( 0, -1) -- (0, 2);
% Red path

\begin{scope}[very thick,decoration={
    markings,
    mark=at position 0.5 with {\arrow{>}}}
    ] 
    \draw[red,postaction={decorate}] (0,0)--(1,0) node[black, midway, below]{\tiny A};
    \draw[red,postaction={decorate}] (1,0)--(1,1) node[black, midway, right]{\tiny B};
    \draw[red,postaction={decorate}] (1,1)--(0,1) node[black, midway, above]{\tiny C};
    \draw[red,postaction={decorate}] (0,1)--(0,0) node[black, midway,  left]{\tiny D};
\end{scope}
% The labels
\node at ( 1.70, -0.2){$x$ };
\node at (-0.24,  1.7){$iy$};

\end{tikzpicture}
\caption{A square contour integral \label{square} on the complex plane}
\end{figure}

<<>>=
integrate.segments(exp, c(0, 1, 1+1i, 1i), close=TRUE)
@ 

Above we see that the result is zero (to within numerical precision),
in agreement with the integral theorem.  It is interesting to consider
each leg separately.  We have

$$
A=e-1\qquad B=e(e^i-1)\qquad C=-e^i(e-1)\qquad D=-(e^i-1)
$$

And taking B as an example:

<<>>=
analytic <- exp(1)*(exp(1i)-1)
numeric  <- integrate.segments(exp, c(1, 1+1i), close=FALSE)
c(analytic=analytic, numeric=numeric, difference=analytic-numeric)
@ 

showing agreement to within numerical precision.

\subsection{The residue theorem}

{\bf residue theorem}.  Given $U$, a simply connected open subset of
$\mathbb{C}$, and a finite list of points $a_1,\ldots,a_n$.
Suppose $f(z)$ is holomorphic on $U_0=U\setminus\left\lbrace
a_1,\ldots a_n\right\rbrace$ and $\gamma$ is a closed rectifiable
curve in $U_0$.  Then

$$\oint_\gamma f(z)\,dz = 2\pi i\sum_{k=1}^n
\operatorname{I}(\gamma,a_k)\cdot\operatorname{Res}(f,a_k)$$

where $\operatorname{I}(\gamma,a_k)$ is the winding number of $\gamma$
about $a_k$ and $\operatorname{Res}(f,a_k)$ is the residue of $f$ at
$a_k$.  \\

The canonical, and simplest, application of this is to derive the log
function by integrating $f(z)=1/z$ along the unit circle, as per
figure \ref{circular}.  Here the residue at the origin is 1, so the
integral round the unit circle is, analytically, $2\pi i$.
Numerically:

<<>>=
u     <- function(x){exp(pi*2i*x)}
udash <- function(x){pi*2i * exp(pi*2i*x)}

analytic <- pi*2i
numeric  <- integrate.contour(function(z){1/z}, u, udash)
c(analytic=analytic, numeric=numeric, difference=analytic-numeric)
@ 



\begin{figure}\centering
\begin{tikzpicture}
% Axes
\draw [help lines,->] (-2,  0) -- (2, 0);
\draw [help lines,->] ( 0, -2) -- (0, 2);

% Red path

\draw [
  decoration = {markings, mark = at position 0 with {\arrow{>}}},
  postaction = {decorate},
  very thick, red] (0,0) circle (1cm);

% The labels
\node at ( 1.70, -0.2){$x$ };
\node at (-0.24,  1.7){$iy$};

\end{tikzpicture}
\caption{A circular contour integral \label{circular} on the complex plane}
\end{figure}


again we see very close agreement.

\subsection{Cauchy's integral formula}

{\bf Cauchy's integral formula}.  If $f(z)$ is analytic within and on
a simple closed curve $C$ (assumed to be oriented anticlockwise)
inside a simply-connected domain, and if $z_0$ is any point inside
$C$, then

$$f(z_0)=\frac{1}{2\pi i}\int_C\frac{f(z)\,dz}{z-z_0}.$$


We may use this to evaluate the Gauss hypergeometric function at a
critical point.  The Gauss hypergeometric function ${}_2F_1(a,b;c;z)$
is defined as

$$1+\frac{ab}{c}\frac{z}{1!} + \frac{a(a+1)b(b+1)}{c(c+1)}\frac{z^2}{2!}+\cdots$$

Now, this series has a radius of convergence of
1~\citep{abramowitz1965}; but the function is defined over the whole
complex plane by analytic continuation~\citep{buhring1987}.  The
\pkg{hypergeo} package~\citep{hankin2015} evaluates ${}_2F_1(a,b;c;z)$
for different values of $z$ by applying a sequence of transformations
to reduce $\left|z\right|$ to its minimum value; however, this process
is ineffective for $z=\frac{1}{2}\pm i\sqrt{3}/2$, these points
transforming to themselves.  Numerically:

<<showhypergeofail>>=
library("hypergeo")
z0 <- 1/2 + sqrt(3)/2i
f <- function(z){hypergeo_powerseries(1/2, 1/3, 1/5, z)}
f(z0)
@

Above we see {\tt NA}, signifying failure to converge.  However, the
residue theorem may be used to evaluate ${}_2F_1$ at this point:

<<>>=
r <- 0.1 # radius of contour
u <- function(x){z0 + r*exp(pi * 2i * x)}
udash <- function(x){r * pi * (0+2i) * exp(pi * 2i * x)}
(val_residue <- integrate.contour(function(z){f(z) / (z-z0)}, u, udash) / (pi*2i))
@ 

We can compare this value with that obtained by a more sophisticated
[and computationally expensive] method, that of
Gosper~\citep{hankin2015}:

<<>>=
(val_gosper <- hypergeo_gosper(1/2, 1/3, 1/5, z0))
abs(val_gosper - val_residue)
@ 

Above we see reasonable numerical agreement.


\section{Conclusions}

The \pkg{elliptic} package includes a suite of functionality for
complex integration using numerical methods and these are used to
demonstrate Cauchy's integral theorem, the residue theorem, and
Cauchy's integral formula.  Numerical errors are generally small.

\bibliography{elliptic}
\end{document}