Computing the residues of a function with a single pole.

Question

Need help with a solution to this problem:

Consider $f : \mathbb{C}\to\mathbb{C}$ with a single pole at point $z = 0$ and analytical elsewhere. Let $f$ be even, so that $f(z) = f(−z)$ for all $z \in \mathbb{C}$. Deduce $\text{Res}(f,0).$

Any advice on where to start?

Answer 1

Let $$f(z) = \sum_{m=-\infty}^{+\infty} a_m z^m$$ on $\mathbb{C}\backslash \{0\}.$ You want to compute $a_{-1}$. Since your function is even, you get $$\sum_{m=-\infty}^{+\infty} a_m z^m = \sum_{m=-\infty}^{+\infty} (-1)^ma_m z^m$$ and hence $$a_{-1} = (-1)^{-1} a_{-1}.$$ You can conclude easily that $a_{-1}=0.$