May 2024
Intermediate to advanced
481 pages
17h 15m
English
Content preview from Complex Analysis
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“ca: ch_04” — 2012/3/27 — 21:16 — page 151 — #40
Complex Integration
|
151
8. If C is closed contour
|
z
|
= r and n =−1, then show that
C
z
n
dz = 0.
9. Let f
(
z
)
=
z
2
+ 5z + 6
z − 2
. Is Cauchy–Goursat theorem applicable to evaluate
C
f
(
z
)
dz if
(a) C is the circle of radius 3 with origin as centre, and
(b) C is a unit circle centred at origin?
Explain your answer in each case.
10. Show that if C is a positively oriented simple closed contour, then the area of the region enclosed
by C can be written as
1
2i
C
¯zdz.
11. Prove that
|
z
|
=1
a
z
dz = 0 for any single-valued branch of a
z
.
12. Use the Cauchy–Goursat theorem for multiply connected domains to evaluate the following ...
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