PROBLEMS FOR HOMEWORK 3
1. Problem 1
Calculate the integral:
|z|= 2
n
z + 1
sin(n2z2)
dz.
2. Problem 2
Using the residue calculus, evaluate the integral
∞
−∞
x sin(nx)
x2 + 2x + 2
dx
(for n even), or the integral
∞
−∞
x cos(nx)
x2 − 2x + 2
dx
(for n odd).
3. Problem 3
Using the residue calculus, evaluate the integral
∞
0
x + n
3
√
x(x2 + 1)
dx.
4. Problem 4
Using the residue calculus, evaluate the integral
∞
0
ln x
k
√
x(x2 − 1)
dx, k = n + 2.
5. Problem 5
Using the residue calculus, evaluate the integral
|z|=n+100
1
nz + 1
cos
1
(z − 1)(z − 2) · · · (z − 99 − n)
dz.
6. Problem 6
Using the residue calculus, find the sum of the series
∞
k=n+1
k2 + n
k4 − n2k2
Problems 1,2 count for 1 point each, problems 3-6 count for 2 points each.
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