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\centerline{{\it  Class 28} {\bf (Friday April 3)}}
 
\begin{description}
\item[SUMMARY] Poles, Zeroes, Residues and Cauchy's Residue Theorem
\vspace{-12pt}
\item[CURRENT READING] Brown \& Curchill pages 193-196
\vspace{-12pt}
\item[NEXT READING] Brown \& Curchill pages 180-196
\end{description}

{\bf Zeroes and Poles}\\
So, we have had a lot of experience finding the {\bf poles} of a function.
The problem of finding a pole is equivalent to finding the {\bf zero} of a
related function. Let's formalize these definitions:

{\bf Zero}\\
A point $z_0$ is called a {\bf zero of order $m$} for the function $f(z)$ if $f$ is analytic at $z_0$ and $f$ and its first $m-1$ derivatives vanish at $z_0$, but $f^{(m)}(z_0) \neq 0.$

{\bf Pole}\\
A point $z_0$ is called a {\bf pole of order $m$} of $f(z)$ if $1/f$ has a zero of order $m$ at $z_0$.

{\bf Identifying Poles and Zeroes}\\
Let $f$ be analytic. Then $f$ has a {\bf zero of order $m$}at $z_0$ if and only if $f(z)$ can be written as ${\displaystyle f(z) = g(z)(z-z_0)^m}$ where $g$ is analytic at $z_0$ and $g(z_0) \neq 0$.\\

If $f(z)$ can be written as ${\displaystyle f(z) = \frac{g(z)}{(z-z_0)^{m}} }$ where $g(z)$ is analytic at $z_0$, then $f$ has a {\bf pole of order $m$} at $z=z_0$ and $g(z_0) \neq 0$\\

How do we find poles of a function? Well, if we have a quotient function $f(z) = p(z)/q(z)$ where $p(z)$ and $q(z)$ are analytic at $z_0$ and $p(z_0) \neq 0$ then $f(z)$ has a pole of order $m$ if and only if $q(z)$ has a zero of order $m$.

\underline{\bf Exercise}\\
We will classify all the singularities of ${\displaystyle f(z) = \frac{3z+2}{z^2+1}}$
\vfill
\underline{\bf Groupwork}\\
Let's try and classify all the singularities of the following functions:
\begin{description}
\item[(a)] $A(z) = {\displaystyle \frac{4}{z^2(z-1)^3}}$
\item[(b)] $B(z) = {\displaystyle \frac{\sin z}{z^2-4}}$
\item[(c)] $C(z) = {\displaystyle \tan z}$
\item[(d)] $D(z) = {\displaystyle \frac{z}{z^2-6z+10}}$
\end{description}


\newpage 

\underline{\bf Residues}\\
Once we know all the singularities of a function it is useful to compute the residues of that function. If a function $f(z)$ has a pole of order $m$ at $z_0$, the residue, denoted by {\bf Res($f;z_0)$} or {\bf Res($z_0$)} is given by the formula below:\\
$$\mbox{\bf Res}(f; z_0) = {\displaystyle \lim_{z \rightarrow {z_0}} \lbrace \frac{1}{(m-1)!}
\frac{d^{m-1}}{dz^{m-1}} \lbrack (z-z_0)^m f(z) \rbrack \rbrace}$$
Find the {\bf residues} of all the singularities we previously classified.\\
{\bf (a)}
\vfill

{\bf (b)}
\vfill

{\bf (c)}
\vfill

{\bf (d)}
\vfill

\underline{\bf Cauchy's Residue Theorem}\\
If $f$ is analytic on a simple (positively oriented) closed contour $\Gamma$ and everywhere inside $\Gamma$ except the finite number of points $z_1, z_2, \cdots z_n$ inside $\Gamma$, then
$${\displaystyle \oint_{\Gamma} f(z) \, dz = 2 \pi i \sum_{k=1}^n \mbox{\bf Res}(f; z_k)}$$

Use Cauchy Residue Theorem to evaluate the following integrals:
\begin{description}
\item[(a)] ${\displaystyle \oint_{|z|=2} \frac{3z+2}{z^2+1}}$
\item[(b)] ${\displaystyle \oint_{|z|=5} \frac{4}{z^2(z-1)^3} \, dz}$
\item[(c)] ${\displaystyle \oint_{|z|=5 \pi} \frac{\sin z}{z^2-4}} \, dz$
\item[(d)] ${\displaystyle \oint_{|z|=2 \pi} \tan z \, dz}$
\item[(e)] ${\displaystyle \oint_{|z|=8} \frac{z}{z^2-6z+10} \, dz}$
\end{description}
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