Math 6720 • Applied Complex Variables and Asymptotic Methods

Complete the following:
\begin{enumerate}
\item
(5 pts) Consider two entire functions with no zeros and having a ratio equal to unity
at infinity. Use Liouville's Theorem to show that they are in fact the same function.
Justify all your steps, including satisfying the assumptions of Liouville's theorem.
\item
(5 pts) Let \(f(z)\) be analytic in and on a circle \(C\) with center \(w\) and radius \(\rho>0\).
From the Cauchy Integral Formula show that
\[
f(w)=\frac{1}{2\pi}\int_0^{2\pi} f\!\left(w+\rho e^{i\theta}\right)\,d\theta.
\]
\end{enumerate}