M417
Fall 1996
hw9.tex due Oct. 30, 1996
Let C = {|z|=1}, traversed counterclockwise. This exercise treats integrals of the form
I(z) =
1
2πi
⌠
(⎜)
⌡
C
h(ζ)
ζ−z
dζ, ζ ∉ C,
for various choices of h(ζ).
For each choice of h(ζ), find
a)
I(z), z inside C
b)
I(z), z outside C
c)
lim
r ↑ 1
−
I(r z
0
), z
0
= e
iθ
0
∈ C
Choices of h(ζ):
0.
h(ζ) = h
0
(ζ) = 1 = (ζ
0
)
n.
h(ζ) = h
n
(ζ) = (ζ
n
), n=1,2,…
-1.
h(ζ) = h
−1
(ζ) = [1/(ζ)] = (ζ
−1
)
-n.
h(ζ) = h
−n
(ζ) = ζ
−n
= ([1/(ζ
n
)]), n=1,2,…
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On 06 Feb 2017, 13:20.