1550251515-Classical_Complex_Analysis__Gonzalez_

462 Chapter 7

-1or0, so that C 1 ,...., C 2 in R but C 1 f C 3 in R. If f is analytic in R, then

J f(z) dz= j J(z) dz
01 02
At this point we may ask whether two contours that are homotopic in
a region R are als~ homologous in R, and conversely. The answer is yes
if R is simply connected, but the converse is not necessarily true if R is
multiply connected. Thus homotopy of contours is a stronger condition
than homology.
We prove:

Theorem 7.19 If C 1 , C 2 are piecewise regular contours with graphs in

a region R, and C 1 is homotopic to C 2 in R, then C 1 is homologous to
C 2 in R, that is,

C1 ~ C2 in R implies C1 ,...., C2 in R

Proof The function 1/(z - a) is analytic in R whenever a rf. R. Since
C 1 ~ C2 in R, by Theorem 7.12 we have

-^1 J ----dz^1 J --dz
27ri z - a 27ri z - a
01 02

i.e., Do 1 (a) = Do 2 (a) or Do 1 -0 2 (a) = 0 for a rf_ R. Hence C1,...., C2 in R.
To show that homology of contours does not imply that the same con-
tours are homotopic in a multiply connected region, consider the following
example (Rudin [34], p. 262). Let ABCD be a square in the complex
plane with diagonals intersecting at M (Fig. 7.15). Let z 1 and z 2 be

interior points of the triangles AM D and BCM, respectively, and let

R = <C - {z1,z2}.

D ~-------~ C

B

Fig. 7.15