Mathematical Methods in the Physical Sciences · Problem 11.36

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · harmonic-mean-value-property

Problem

Show that a harmonic function

u(x, y)

is equal at every point

a

to its average value on any circle centered at

a

[and lying in the region where

f(z) = u(x, y) + iv(x, y)

is analytic]. Hint: In (3.9), let

z = a + re^{i\theta}

(that is,

C

is a circle with center at

a

), and show that the average value of

f(z)

on the circle is

f(a)

(see Chapter 7, Section 4 for discussion of the average of a function). Take real and imaginary parts of

f(a) = [u(x, y) + iv(x, y)]_{z=a}