A First Course in Complex Analysis · Problem 7

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

Mathematics — Complex Analysis · holomorphic-functions

Problem

Prove the following generalization of Theorem 8.1.1: Suppose

(f_n)

is a sequence of functions that are holomorphic in a region

G

, and

(f_n)

converges uniformly to

f

G

. Then

f

is holomorphic in

G

. (This result is called the Weierstra\ss\ convergence theorem.)