1550251515-Classical_Complex_Analysis__Gonzalez_

458 Chapter^7

R c

Fig. 7.13

a denotes some point inside Ci, we can write (7.12-2) in the form

J f(z)dz = no1(a) J f(z)dz = nc(a) J J(z)dz (7.12-3)

c ct ct

where nc(a) = nc1(a) means the index or winding number of c, or C1,

about a (see Section 3.16).

7.13 ANALYTICAL REPRESENTATION OF THE

WINDING NUMBER. APPLICATIONS

The last formula of the preceding section leads to a useful analytical

representation of the winding number no( a). In fact, by applying for-

mula (7.12-3) to the function f(z) = l/(z-a), and letting ct be the circle
z - a = reit, 0 :::; t :::; 27r, we obtain

J

!:! = nc(a) J !:! = nc(a) f

2

"' idt = nc(a). 27fi

z-a z-a h

G ct

so that

1 j dz

no( a)= 27fi z - a (7.13-1)

G

Formula (7.13-1) will now be used to define the winding number of

any closed piecewise regular contour C (not necessarily a simple one) with

respect to an arbitrary point a f/. C*. It is sufficient to consider a piecewise

regular closed contour C, since any rectifiable contour can be approximated