1550251515-Classical_Complex_Analysis__Gonzalez_

452 Chapter^7

In what follows we state as corollaries some particular versions of

the Cauchy-Goursat theorem that are often found in books on complex

analysis.

Corollary 7.12 Let f be analytic in a simply connected region G, and

let C be a closed contour with graph contained in G. Then

J f(z) dz= 0 (7.10-15)

c

Proof If G is simply connected, then C is homotopic to a point in G (Def-

inition· 3.30). Note that the contour C may be described any number of

times in either direction. Also, we observe that C need not be simple. For

instance, for the contours shown in Fig. 7.11, the integral (7.10-15) is still

zero (of course, with f analytic in G), since it vanishes on each "loop" of

the contours, and also along the connecting link PQ, which is described

the same number of times in opposite directions.

Remark The simply connected region G may be taken to be a suitable

subregion of a multiply connected region M.

Corollary 7.13 Let f be analytic on and within the graph of a simple

closed contour C. Then

J f(z) dz= 0

c

Proof The hypothesis of the theorem implies that f is analytic in a simply

connected region G :J C* U Int C*.

Note The conditions stated in Theorem 7.12 and in Corollaries 7.12

and 7.13 are sufficient (but not necessary) for fc f(z) dz to be zero. For

G

Fig. 7.lll