Math 6720 • Applied Complex Variables and Asymptotic Methods

Analytic functions
\begin{enumerate}
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(12 points) Let \(u(x,y)=\dfrac{y}{x^2+y^2}\) be the real part of an analytic function \(f(z)\). Find the
imaginary part \(v(x,y)\) and the function \(f(z)\).

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(13 points) Let \(f(z)\), \(g(z)\) be two entire functions with no zeroes such that
\[\lim_{z\to\infty}\frac{f(z)}{g(z)}=1.\]
Use Liouville's Theorem to show that \(f(z)\equiv g(z)\) for all \(z\).
\end{enumerate}

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