464 Chapter^7
Proof We have
J f(z)dz = t J f(z)dz
G k=l'Yk
where each /k is described according to the direction and number of times
that C winds about ak. ButJ f(z) dz= f!c(ak) J f(z) dz= mk J f(z) dz
'Yk 'Yt 'YtTherefore, (7.13-4) holds.7.14 A Further Extension of the Cauchy-Goursat Theorem
To state the Cauchy-Goursat theorem in its most general form, we need
the following definition.Definition 7 .9 A function f is said to be locally bounded at a point a
if f is analytic in some neighborhood N 0 (a), except possibly at a, and if
lf(z)I < M (a constant) in NHa). Of course, if f is analytic at a, it is
continuous at that point, so its absolute value is bounded in some N 0 (a).
In fact, f(z)-+ f(a) implies that lf(z)I-+ lf(a)I, so corresponding to E = 1there is a 8 > 0 such that for lz - al< 8 we have -1 < lf(z)I - f(a)I < 1,
or IJ(z)I < lf(a)I + 1 = M.
Note that if f is locally bounded at a, then
lim(z - a)f(z) = 0
z-+aTheorem 7 .21 Let R be a region and suppose that f is analytic in R
except at a finite number of points ak (k = 1, ... , n), where f is locally
bounded. If the closed contour C is homologous to zero in R and ak r:f. C*
(k = 1, ... ,n), then
J f(z)dz = 0
G
Proof Let mk = Slc(ak)· By Theorem 7.20 we havej f(z)dz = tmk j f(z)dz
G k=l 'Yk +(7.14-1)