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Math 326
Homework Assignment 6
due date: Oct. 28, 2010
1. Let f (z) = z 2 .
(a) By parameterizing the curve and computing the integral, find
n
o
π
is the curve z |z| = 3, 0 ≤ Arg(z) ≤ 2 .
(b) By parameterizing the curve and computing the integral, find
Z
f (z) dz where γ0
Z
f (z) dz where γ1
γ0
γ1
is the straight line from 3 to 3i.
(c) By finding an antiderivative for f , calculate the integral for any path starting at
3 and ending at 3i.
2. Let z0 ∈ C be a fixed complex number, and for any real number
r > 0 let γr be the
Z
(z − z0 )n dz, where n is
circle of radius r around z0 , oriented counterclockwise. Find
γr
an integer (your answer will depend on n).
Note: You can use the previous homework assignment to take care of the case n = −1.
For the remaining cases you should be able to use some of the theorems we know to
compute the answer without explicitly integrating the function.
3. Let
3
7
5
1
+
+
+
+ sin(ez ),
z − 2i z + 2 − 3i z + 1 z − 3 + 3i
and let γ be the circle of radius 4 centred at z = 0 and oriented counterclockwise.
f (z) =
(a) The function f is described as a sum of five different functions. For each of those
functions state the largest domain on which they are holomorphic.
Z
(b) Use Cauchy’s theorem (and perhaps previous homework results) to find f (z) dz
γ
without parameterizing γ or doing any explicit integration.
Be sure to explain the steps of your argument (i.e., exactly how you are using the
theorems) clearly.
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4. Use partial fractions and Cauchy’s theorem to compute
Z
Z
Z
2
4
2z
dz
(b)
dz
(c)
dz
(a)
2
2
2
|z|=2 z + 1
|z|=2 z − 4z + 3
|z|=2 z + 1
The integrals around the contour |z| = 2 should be taken counterclockwise. Note that
z 2 + 1 factors as (z + i)(z − i) over the complex numbers.
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