A First Course in Complex Analysis · Problem 34

Isolated Singularities and the Residue Theorem — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · laurent-series

Problem

Recall that a function

f : G \to \mathbb{C}

is even if

f(-z) = f(z)

for all

z \in G

, and

f

is odd if

f(-z) = -f(z)

for all

z \in G

. Prove that, if

f

is even (resp., odd), then the Laurent series of

f

0

has only even (resp., odd) powers.