A First Course in Complex Analysis · Problem 30

Examples of Functions — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · constant-functions

Problem

Prove that, if

G \subseteq \mathbb{C}

is a region and

f: G \to \mathbb{C}

is a complex-valued function with

f''(z)

defined and equal to

0

for all

z \in G

, then

f(z) = az + b

for some

a, b \in \mathbb{C}

. Hint. Use Theorem 2.4.2 to show that

f'(z) = a

, and then use Theorem 2.4.2 again for the function

f(z) - az