Page 729 - Advanced_Engineering_Mathematics o'neil

20.3 Consequences of Cauchy’s Theorem 709

f (z) − f (z 0 ) f (z 0 +re ) − f (z 0 )
2π it

it
re
dz = it ire dt
C z − z 0 0
2π

it
( f (z 0 +re ) − f (z 0 ))dt
=
0
2π
it
≤ | f (z 0 +re ) − f (z 0 ) | dt.
0
it
But | f (z 0 +re ) − f (z 0 ) |→ 0as r → 0 by continuity of f . We conclude that

f (z) − f (z 0 )

dz = 0,

C z − z 0
and hence, that
f (z) − f (z 0 )

dz = 0,
C z − z 0
establishing Cauchy’s integral formula.
The integral formula gives added appreciation of the power of the condition of differentiabil-
ity for complex functions. The formula gives the value of f (z) at all points enclosed by a closed
path γ , strictly in terms of values of f (z) at points on γ , because these are all that are needed to
evaluate
f (z)

dz.
γ z − z 0
By contrast, knowing the values of a differentiable real-valued function g(x) at the endpoints of
an interval [a,b] tells us nothing about values g(x) for a < x < b.
Another implication of the integral formula for higher derivatives is that a complex function
that is differentiable on an open set has derivatives of all orders on that set. Again, real functions

do not behave this well. If g (x) exists, g (x) may not.
20.3.4 Properties of Harmonic Functions
As an application of Cauchy’s integral formula, we will derive two important properties of har-
monic functions. This is a prime example of the use of complex functions to derive facts about
real functions—a theme we will see again and which is made possible by the connection between
harmonic functions and the real and imaginary parts of differentiable complex functions.

THEOREM 20.5 The Mean Value Property

Let u be harmonic on a domain D.Let (x 0 , y 0 ) be any point of D, and let C be a circle of radius
r in D centered at (x 0 , y 0 ) and enclosing only points of D. Then
1 2π
u(x 0 , y 0 ) = u(x 0 +r cos(θ), y 0 +r sin(θ))dθ.
2π 0
Notice that (x 0 +r sin(θ), y 0 +r sin(θ)) are polar coordinates of points on the circle, varying
once counterclockwise over the circle as θ varies from 0 to 2π. The theorem therefore says that
the value of u at the center of the circle is the average of the values of u(x, y) over the circle.
Proof For some v, f = u + iv is harmonic on D.Let z 0 = x 0 + iy 0 . By Cauchy’s integral
formula,

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 15:32 THM/NEIL Page-709 27410_20_ch20_p695-714