Homework Assignment 5, MATH 426/526, Spring 09
Denote by C(a;r) the curve whose image is the circle with center aand radius
r > 0, i. e. with parametrization
z(t) = a+rit,0≤t≤2π
Problem 16) (10 pts) Calculate, using the Cauchy integral theorem and the
Cauchy integral formula, the following integrals:
(a) IC(2;1)
z7+ 1
z2(z4+ 1)dz
(b) IC(1; 3
2)
z7+ 1
z2(z4+ 1)dz
(c) IC(0;3)
e−z
(z+ 2)3dz
(d) IC(0;3)
cos(πz)
z2−1dz
Problem 17) (10 pts) Compute, using the Cauchy integral theorem and the
Cauchy integral formula the following integrals:
(a) 1
2πi IC(i;1)
ez
z2+ 1dz
(b) 1
2πi IC(−i;1)
ez
z2+ 1dz
(c) 1
2πi IC(0;3)
ez
z2+ 1dz
(d) 1
2πi IC(1+2i;5)
4z
z2+ 9dz
Additional Assignments 5 for MATH 526
Problem G8) (16 points) Compute
(a) IC(1;1) z
z−1n
dz , n ∈N
(b) IC(0;r)
1
(z−a)n(z−b)mdz , |a|< r < |b|, n, m ∈N
