Complex Analysis
Spring 2001
Homework V Solutions
1. Conway, chapter 4, section 5, problem 7.
Let
γ
(
t
) = 1 +
e
it
for 0
≤
t
≤
2
π.
Find
γ
(
z
z

1
)
n
dz
for all positive integers
n.
By Corollary 5.8, this is
2
πi
(
n

1)!
times the
n

1rst derivative of
f
(
z
) =
z
n
evaluated at
z
= 1
.
The
n

1rst derivative of
z
n
is
n
!
z
and so the result is 2
nπi.
2. Conway, chapter 4, section 5, problem 9. Show that if
f
:
C
→
C
is continuous and
analytic off the interval [

1
,
1] then
f
is entire.
This is an application of Morera’s theorem. One has to show that the integral of
f
over
the boundary of any rectangle is equal to zero. For rectangles which are disjoint from
the interval, this follows from Cauchy’s theorem. Rectangles which meet the interval
can be broken into a union of subrectangles which lie on and above the xaxis and
rectangles which lie on or below the axis. For any such rectangle which intersects the
interval, show that the integral is given as the limit as
δ
→
0 of the integrals over
rectangles where the side lying on the horizontal axis is moved distance
δ
away from
the axis. This is where continuity is used, since if
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 Spring '01
 SNIDER
 Derivative, Integers, Complex number, dz

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