Characterize all entire functions with a pole of order \(m\) at \(\infty\).
Solution
Since \(f\) is entire, \(f(z) = \sum_{k\geq 0 } c_k z^k\). Expanding about \(z_0=\infty\), we have \(f(1/z) = \sum_{k\geq 0} c_k z^{-k} = c_0 + {c_1\over z} + \cdots\). If \(z_0=\infty\) is a pole of order \(m\), then \(c_m\neq 0\) but \(c_{>m} = 0\), which forces \(f(z) = \sum_{0\leq k \leq m} c_k z^k\) to be a polynomial of degree \(m\).