Show that if \(f\) is entire and \(f(1/z)\) has a pole at \(z=0\), then \(f\) is a polynomial.
Solution
Write \(f(z) = \sum_{k\geq 0}c_k z^k\), so \(g(z) \da f(1/z) = \sum_{k\geq 0} c_k z^{-k}\). Since \(z=0\) is a pole of \(g\), \(c_k = 0\) for all \(k\geq m\) for \(m\) the order of the pole, so \(f(z) = \sum_{0\leq k\leq m}c_k z^k\) is a polynomial of degree at most \(m\).