Find all entire functions with have poles at \(\infty\).
Solution
Goal: Find all entire functions \(f\) such that \(f\) has a pole at \(\infty\).
1. A pole at \(\infty\) means \(g(w) \definedas f(1/w)\) has a pole at \(w = 0\).
This is the definition of the singularity at \(\infty\): \(f\) has a pole at \(\infty\) iff \(f(1/w)\) has a pole at \(0\).
2. If \(f\) is entire with a pole at \(\infty\), then \(f\) is a nonconstant polynomial.
By 1, \(g(w) = f(1/w)\) has a pole at \(w=0\), so its Laurent expansion about \(0\) has only finitely many negative terms: \(g(w) = \sum_{n=-N}^{\infty} b_n w^n\) with \(b_{-N} \neq 0\).
Substituting \(w = 1/z\) gives \(f(z) = \sum_{n=-N}^{\infty} b_n z^{-n} = \sum_{k=-\infty}^{N} b_{-k} z^k\); since \(f\) is entire (no poles in \(\CC\)), all coefficients of negative powers vanish and \(f(z) = \sum_{k=0}^{N} a_k z^k\) is a polynomial, nonconstant because the pole at \(\infty\) has positive order (\(N \geq 1\)).
3. Conversely, every nonconstant polynomial has a pole at \(\infty\).
If \(f(z) = a_N z^N + \cdots + a_0\) with \(N \geq 1\) and \(a_N \neq 0\), then \(f(1/w) = a_N w^{-N} + \cdots + a_0\) has a pole of order \(N\) at \(w = 0\), so \(f\) has a pole of order \(N\) at \(\infty\) by 1.
4. Q.E.D.
2 and 3 together show the entire functions with a pole at \(\infty\) are exactly the nonconstant polynomials.