A First Course in Complex Analysis · Problem 8.4.11

Taylor and Laurent Series — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis

Problem

In the proof of Theorem 8.2.2, prove that

X

and

Y

are open sets, where

X := \{a \in G : \text{there exists } r \text{ such that } f(z) = 0 \text{ for all } z \in D[a, r]\}

and

Y := \{a \in G : \text{there exists } r \text{ such that } f(z) \neq 0 \text{ for all } z \in D[a, r] \setminus \{a\}\}