1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 463

Consider the closed contour C 1 = ADBCDACBA. Since in every inte-

gral along C 1 the integral over AD cancels with the integral over DA, and

similarly, the integrals over BC and CB cancel each other, C 1 is homolo-

gous to C2 = ACDBA. But z1 and z 2 are in the unbounded component

of ( C;)', so that C 2 "' 0 in R, and hence C 1 "' 0 in R also. However, C 1

is not homotopic to a point in R, i.e., C 1 cannot be continuously deformed

to a point within R.

Theorem 7 .20 Suppose that f is analytic in some region R, except pos-

sibly at the points ak ( k = 1, ... , n ), and that the closed contour C is

homologous to zero in R and winds mk times about ak. If ak fj. C*

( k = 1, ... , n), then

j f(z)dz = tmk j f(z)dz

c k=l 'Yk + ' '

_(7.13-4)

where 'Yt is a circle about ak described once in the positive direction with

sufficiently small radius so that 'Yk n C* ·= 0 and 'Yk n 'Yj = 0 (Fig. 7.16),

where k -=/= j.

Note In this theorem, as well as in what follows, the contours we refer
to may be piecewise regular or, more generally, rectifiable, in view of our
discussion of the winding number in Section 3.16 and the observations
made at the beginning of this section.

Fig. 7.16