1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 461

following manner: If for any point a E Int C* we have ilo( a) = + 1 the
contour C is described once in the positive direction, and if ilo(a) = -1,
then C is described once in the negative direction. The contours C and
-C are described in opposite directions. Also, we may characterize the

notions of interior and exterior of a closed contour as follows: If a tj C

and ilo( a) = k -=f O, a is an interior point of C*; if ilo( a) = O, then a
is an exterior point.
We recall (Definition 3.31) that a cycle C, in particular a closed contour,
with graph contained in an open set A is said to be homologous to zero
in A, written C,...., 0 in A, if for every point z 0 tj A we have ilo(z 0 ) = 0.
Also, we say that two chains, in particular two cycles, are homologous in
A, denoted C1 ,...., C2, if C1 + (-C2) = C1 - C2 ,...., 0 in A.
A homological proof, using the concept of winding number, can also
be given of the Cauchy-Goursat theorem. For this we refer the reader to
Ahlfors [1], to Ash [3], or to Rudin [34], where the J. D. Dixon proof [10]

is presented. Thus if f is analytic in a region R and if C 1 ,...., C2 in R, then

j f(z) dz= j J(z) dz

01 02

Example Consider the doubly connected region R bounded by r and r',

and the closed contours C 1 , C 2 , C 3 (Fig. 7.14) in R. With respect to a
point a tj R, we have ilo 1 (a) = ilo 2 (a) = 0, and, assuming C3 as described
once in the positive direction, we have !10 3 (a) = 1 or 0, depending on the
component of R' in which a lies. Hence !10 1 -o 2 (a) = 0 but !10 1 -o 3 (a) =

a'•

r

Fig. 7.14