1550251515-Classical_Complex_Analysis__Gonzalez_

528 Chapter 8

so that under such conditions the reversal of the order of integrations is

permissible.

In fact, if z 0 is a fixed point and z a variable point in G, the integral

F(z,t) = r f((,t)d(
lzo
is independent of the path of integration ( G being simply connected) and so,
for fixed t, represents an analytic function for z E G. In addition, for fixed
z, F(z, t) is continuous with respect to t E R+, and IF(z, t)I is bounded
for z on any compact subset of G and t E R+. Then, by Theorem 7.37,
the derivative of the integral

1T F(z,t)dt = 1T [1: f((,t)d(] dt

with respect to z can be obtained by differentiation under the integral
sign, i.e.,

8 1T {T {) 1T

az

0

F(z, t) dt =Jo oz F(z, t) dt =

0

f(z, t) dt (8.2-4)

Also, we have

! 1: [1T f((,t)dt] d( = 1T f(z,t)dt (8.2-5)

Equations (8.2-4) and (8.2-5) imply that

(8.2-6)

since both functions have the same derivative and both reduce to zero for
Z = Zo.
Now because ft' f((, t) dt converges uniformly on every compact subset
of G and, in particular, on any contour C with graph in G, joining z 0 and
z, it follows that for any given f there is a T 0 such that for T 2:: To we
have, in view of (8.2-6),

j [J(°

0

J((,t)dt] d(-1r [fct((,t)d(] dt

c

= J [1

00

f((,t)dt] d(-J [1T f((,t)dt] d(

c c