Integration 451= J f(z)dz (7.10-12)
h(8S)
Hence, by Corollary 7.11, we get, in view of (7.10-11) and (7.10-12),J f(z) dz= 0
h(8S)
This takes care of part 1 of Lemma 7.4. As to part 2, it suffices to
note that under the assumption h(O,r) = h(l,r), the second sum on the
right-hand side of (7.10-12) cancels with the fourth, to give
J f ( z) dz - J f ( z) dz = J f ( z) dz + J f ( z) dz = 0
h(8SB) -h(8ST) h(8SB) h(8ST)Theorem 7.12 (Joint Homotopy Version of the Cauchy-Goursat Theo-
rem). Let f be analytic in a simply or multiply connected region G, and
let C 1 and C2 be two contours with graphs in G. Then
J f(z) dz= J f(z) dz (7.10-13)
01 02if either
l. C 1 and C 2 are fixed endpoints homotopic in G, or
- C 1 and C 2 are closed and homotopic (freely homotopic) in G. In
particular, we have
j f(z)dz = 0 (7.10-14)
01
if C 1 is homotopic to a point in G.Proof All that is needed now is to speciali~e the results in Lemma 7.4.
Assume that h(8SB) = C1: z = z1(t), 0 :5 t :5 1 and -h(8ST) =
C 2 : z = z 2 (t), 0 :5 t :5 1 (after reparametrization, if necessary), A = G.
Then h: S-+ G defines a homotopy of C1 into C 2 in G, and part 1 of the
theorem is the special case of part 1 of the lemma in which the left-and
right-hand sides of S map each into a point. Similarly; part 2 of the theorem
is the same as part 2 of the lemma with the additional assumption that
z 1 (0) = z 1 (1), z 2 (0) = z 2 (1), i.e., that the contours C1 and C2 are closed.
Of course, if C 2 reduces to a point, then f 02 f(z) dz = 0 and (7.10-14)
follows from (7.10-13)..