= Polynomial growth forces a meromorphic function to be rational
A <meromorphic function> on the plane with $|f(z)|\leq C|z|^n$ outside a disc has no poles there and only finitely many poles inside. Subtract their finite <principal parts> to obtain an <entire function>. For $n\geq0$, <Cauchy estimates> make every derivative of order above $n$ vanish, so the remainder is a polynomial. For $n<0$ the remainder tends to zero at infinity and is zero by <Liouville theorem>. Adding back the principal parts gives a <rational function> in either case.
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