Mathematical Methods in the Physical Sciences · Problem 11.3

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · liouville-theorem

Problem

Liouville's theorem: Suppose

f(z)

is analytic for all

z

(except

\infty

), and bounded [that is,

|f(z)| \leq M

for all

z

and some

M

]. Prove that

f(z)

is a constant. Hints: If

f'(z) = 0

, then

f(z) = \text{const.}

To show this, write

f'(z)

as in Problem 3.21 where

C

is a circle of radius

R

and center

z

, that is,

w = z + Re^{i\theta}

. Show that

|f'(z)| \leq M/R

, and let

R \to \infty