1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 473

10. Study the multiple-valued function defined by

F( ) = ("Y) 1z _!:L

z 0 1 + (2

in C - {i, -i}, and show that F(z) = arctanz.

7.17 Cauchy's Integral Formula

The following basic result, also due to Cauchy, shows that the values of an

analytic function inside a simple closed contour are completely determined

by the values assumed by the function on the contour.

Theorem 7 .23 If f is analytic in a simply connected region R containing

the graph of the simple closed contour C, then

Uc(z)f(z) = -

2

1

. J f~() d(

7ri - z

(7.17-1)

c

for any point z such that z ¢ C*.

Proof First suppose that z E Int C. Given € > 0 there exists r > 0 such
that the disk I( - zl:::; r lies in Int C and for which If(()-f(z)I < €. Let
C 1 be a circle with center z and radius r described as many times and in
the same direction as C (Fig. 7.20).

Since f ( () / ( ( - z), as a function of (, is analytic on both C* and Ci,

as well as on the doubly connected region bounded by those curves, we

....

R

Fig. 7.20