Find all entire functions with have poles at \(\infty\).
Solution
If \(f\) is entire, write \(f(z) = \sum_{k\geq 0}c_k z^k\) and \(g(z) \da f(1/z) = \sum_{k\geq 0}c_k z^{-k}\). If \(z=\infty\) is a pole of order \(m\) of \(f\), \(z=0\) is a pole of order \(m\) of \(g\), so \[ g(z) = \sum_{0\leq k \leq m}c_k z^{-k} \implies f(z) = \sum_{0\leq k \leq m}c_k z^k ,\] making \(f\) a polynomial of degree at most \(m\).