Math 6720 • Applied Complex Variables and Asymptotic Methods

\begin{enumerate}
\item
(8 points) Let \(P(z)\) be a polynomial of degree \(n\), with \(n\) simple roots, none of which lie
on a simple closed contour \(C\). Evaluate
\[
I=\frac{1}{2\pi i}\oint_C \frac{P'(z)}{P(z)}\,dz.
\]
\item
(8 points) Use Liouville's Theorem to prove the Fundamental Theorem of Algebra. Let
\(P(z)\) be any polynomial of integer degree \(m\ge 1\). Then there is at least one point \(z=\alpha\)
such that \(P(\alpha)=0\); that is \(P(z)\) has at least one root.
\item
(9 points) Let \(f(z)\) be a meromorphic function, i.e. a function that only has poles in the
finite plane, defined inside and on a simple closed contour \(C\), with no zeros or poles on \(C\).
Let \(N\) and \(P\) be the number of zeroes and poles, respectively, of \(f(z)\) inside \(C\); where a
multiple zero or pole is counted according to its multiplicity. Show that
\[
I=\frac{1}{2\pi i}\oint_C \frac{f'(z)}{f(z)}\,dz=N-P.
\]
\end{enumerate}

\item