3.II.14A

State the Cauchy integral formula, and use it to deduce Liouville's theorem.

Let $f$ be a meromorphic function on the complex plane such that $\left|f(z) / z^{n}\right|$ is bounded outside some disc (for some fixed integer $n$ ). By considering Laurent expansions, or otherwise, show that $f$ is a rational function in $z$.

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