Integration 455Remark If the region R does not contain a point b such that every ray
with origin b meets C at just one point, it can often be decomposed into
subregions having this property. Since the integral along the boundary
of each subregion is then zero, the integral will also be zero along the
boundary of R.7.12 Cauchy-Goursat Theorem for Several Contours
Theorem 7 .15 Let f be analytic in a region G and consider n + 1 simple
closed contours C, C1, ... , Cn with graphs contained in G and satisfying
the following conditions:- C'k c Int C* ( k = 1, ... , n)
2. C'k c Ext CJ (k,j = 1, ... ,n;k-:/= j)
- G ::J R, where R is the multiply connected region
Int C* - Int C{ - .. · - Int C~
with boundary f = C UC~ U ···UC~. Then
J f(z)dz= J f(z)dz+ .. ·+ J f(z)dz
C C1 Cn(7.12-1)where all the contours are described the same number of times in either
the positive or the negative direction.Proof If G is simply connected the property is obviously true, since in
this case each of the integrals in (7.12-1) reduces to zero. Suppose that
G is multiply connected, and to fix ideas, consider the case n = 2, as
illustrated in Fig. 7.12. Now we join point A of the contour C to point Bof C 1 by a rectifiable simple arc AB. If it happens, as in Fig. 7.12; that
parts of the graph of AB do not lie in R, all we need to do is to replacethe point A by the last point of intersection A' of AB with the curve C
(as we describe the arc AB from A to B). Then the graph of the arc
A' B = /l will be contained in R. Similarly, we join C1 to C2 by a simple
rectifiable arc DE= 12 with graph contained in R, and C 2 to C by an arc
F H = ')' 3 satisfying the same conditions. We choose the connecting arcs/l, 1' 2 , ')' 3 neither with points in common nor with points in common with
the contours C, C 1 , C 2 , excepting the corresponding endpoints.It is sufficient to consider the contours C, C1, C2 as described once in
the positive direction. Clearly, if (7.12-1) holds in this case, it will also
be true when all orientations are reversed, and also when all of them are
described k times in either direction.