Page 413 - Schaum's Outline of Theory and Problems of Advanced Calculus

404 FUNCTIONS OF A COMPLEX VARIABLE [CHAP. 16

ð
P 2
(b) Under these conditions prove that f ðzÞ dz is independent of the path joining P 1 and P 2 .
P 1
þ þ þ þ
udx vdy þ i vdx þ udy
ðaÞ f ðzÞ dz ¼ ðu þ ivÞðdx þ idyÞ¼
C C C C
By Green’s theorem (Chapter 10),
ðð ðð
þ þ
@v @u @u @v
dx dy; dx dy
udx vdy ¼ vdx þ udy ¼
C @x @y C @x @y
r r
where r is the region (simply-connected) bounded by C.
@u @v @v @u
Since f ðzÞ is analytic, ¼ ; ¼ (Problem 16.7), and so the above integrals are zero.
@x @y @x @y
þ
Then f ðzÞ dz ¼ 0, assuming f ðzÞ [and thus the partial derivatives] to be continuous.
0
C
(b) Consider any two paths joining points P 1 and P 2 (see Fig. 16-5). By Cauchy’s theorem,
ð
f ðzÞ dz ¼ 0
P 1 AP 2 BP 1
P 2
ð ð
Then f ðzÞ dz ¼ 0
Path 1
f ðzÞ dz þ
P 1 AP 2 P 2 BP 1
A
ð ð ð
B
or f ðzÞ dz ¼ f ðzÞ dz ¼ f ðzÞ dz
Path 2
P 1 AP 2 P 2 BP 1 P 1 BP 2 P 1
i.e., the integral along P 1 AP 2 (path 1) ¼ integral along P 1 BP 2
Fig. 16-5
(path 2), and so the integral is independent of the path joining P 1
and P 2 .
2
This explains the results of Problem 16.10, since f ðzÞ¼ z is analytic.
16.12. If f ðzÞ is analytic within and on the boundary of a region bounded by two closed curves C 1 and C 2
(see Fig. 16-6), prove that
þ þ
f ðzÞ dz
f ðzÞ dz ¼
C 1 C 2
As in Fig. 16-6, construct line AB (called a cross-cut)connecting any point on C 2 and a point on C 1 .By
Cauchy’s theorem (Problem 16.11),
ð
f ðzÞ dz ¼ 0
AQPABRSTBA
since f ðzÞ is analytic within the region shaded and also on the
boundary. Then
ð ð ð ð
f ðzÞ dz ¼ 0
f ðzÞ dz þ f ðzÞ dz þ f ðzÞ dz þ ð1Þ
AQPA AB BRSTB BA
ð ð
But f ðzÞ dz ¼ f ðzÞ dz. Hence, (1)gives
Fig. 16-6
AB BA
ð ð ð
f ðzÞ dz
f ðzÞ dz ¼ f ðzÞ dz ¼
AQPA BRSTB BTSRB