678 Chapter 9
as before.
Theorem. 9.10 (General Form of the Residue Theorem). Suppose that
the function f is analytic in some open set A except for isolated singularities
ak ( k = 1, ... , n). Let C be a closed contour homotetic to a point in A and
not passing through any of the singular points ak. Then
j f(z) dz= 27ri t ilc(ak) Jl~~ f(z)
c k=l(9.9-5)the contour C being described a given number of times in either direction.
Proof As in Theorem 9.9, consider the circles ct: z - ak = rkeit, 0 $
t $ 271", with radii rk, small enough so that CZ n CJ = 0 (k -=/:-j) and
Ck n C* = 0 (Fig. 9.8). By Theorem 7.20 and Definition 9.11 we have
1 J 1 n J
2 7ri f(z) dz= 2 7ri L ilc(ak) f(z) dz
c ~1 ck +n 1 J n
= L ilc(ak)-
2. f(z) dz='°' ilc(ak) Res f(z)
71"t L.J z=ak
k=l + k=l
ck
Note Cauchy's formula (7.17-1) is a special case of (9.9-5) since the
residue of f(z)/(z -a), where f is regular in A, is f(a).
Fig. 9.8