1550251515-Classical_Complex_Analysis__Gonzalez_

476 Chapter^7

(;

2 3 x

c

Fig. 7.21

=^2 7ri 2i(i i_ 3) +^2 7ri -2i( =;-3) = - ~ 7ri

The following are generalizations of Theorem 7.23.

Theorem 7.24 Let R be a multiply connected region bounded by n + 1
simple closed contours C, 01 , ••. , Cn all described the same number of
times in the same direction, and such that

1. Ci; c Int C* ( k = 1, ... , n)

2. Ci; c Ext CJ (k,j = 1, ... , n; k -=/= j)

Let r = C + ( -0 1 ) + · · · + ( -Cn) be called the total boundary of R,

and suppose that f is analytic on fl, except possibly at a finite number of

points ar E R (r = 1, ... ,p) where f is locally bounded. Then for any z

such that z <:/. r*' z -=/= ar, we have

r!r(z)f(z) = ~ j J(() d(

27ri ( - z

(7.17-5)

r

Proof Let 'Y be a small circle about z, and 'Yr a small circle about ar

(r = 1, ... ,p), with radii small enough so that all circles are contained in

R, each being exterior to all others (Fig. 7.22).
By Theorem 7.15 we have

___!__ J f(()d( = r!r(z) J J(()d( + t r!r(ar) 1 J(()d(

27ri ( - z 27ri ( - z 27ri -y+ ( - z

C -y+ r=l r

+ ~ ~ J J(()d(

~ 27ri ck (-z