A First Course in Complex Analysis · Problem 23

Examples of Functions — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · holomorphic-functions

Problem

Suppose

f

is entire and can be written as

f(z) = u(x) + i\,v(y)

, that is, the real part of

f

depends only on

x = \operatorname{Re}(z)

and the imaginary part of

f

depends only on

y = \operatorname{Im}(z)

. Prove that

f(z) = az + b

for some

a \in \mathbb{R}

and

b \in \mathbb{C}