446 Chapter^7which shows that Iimh-+o[F(z + h) - F(z)]/h = J(z).
Remark Obviously, in Lemma 7.2 the disk lz -z 01 < r can be replaced
by any region that contains the rectangle with opposite vertices Zo and Z
as long as it contains the point z. Such are the interior of a rectangle or
a half-plane. More generally, Lemma 7.2 is valid on any convex open set.
See Exercises 7.2, problem 9.Corollairy 7 .9 If f is analytic in the disk lz -z 0 I < r then
J f(z) dz= 0
cfor any dosed c~ntour C with graph contained in the disk.
Proof It follows from Lemma 7.2 and Corollary 7.4.
Corollary 7 .10 If f is analytic in the complex plane (of in half of the
complex plane), thenJ f(z) dz= 0
cfor any closed contour C with graph contained in C (or in that half of the
complex plane).
Proof Since the graph of C is compact, and so a bounded set, it is contained
in a disk with center at the origin and sufficiently large radius. Hence the
case off analytic in C follows from Corollary 7.9. For the case off analytic
in half of the complex plane it suffices to take into account the remark
following the proof of Lemma 7.2 together with Corollary 7.4.
As already noted, the above are special cases of the Cauchy-Goursat
theorem. To obtain the theorem with full generality we proceed as follows.
First, let G be any region and suppose that the graph of the rectifiable
arc 1: z = z(t), a::; t::; (3 is contained in G (Fig. 7.9). Let p = d(1*,G'),
where G' = C-G. By Exercises 3.3, problem 5(b), p > 0. If we choose e
such that 0 < e < p, by the uniform continuity of z(t) in [a, (3] there is a
8 > 0 such that t, t' E [a, (3] and it -t'I < 8 implies that lz(t) -z(t')I < e.
Hence ifa = to ::; ti ::; t2 · · · ::; tn = (3
gives a partition of the interval [a, (3] with norm less than 8, the image of
each subinterval [tk, tk+i] (k = 0, 1, ... , n - 1) under z = z(t) lies in a disk
Dk CG, with a center at Zk = z(tk) and radius e. Now if f is analytic in