Math 210B, Winter 2010
Homework Assignment 3
Due Monday, February 17, 2010
1. Compute the integral
ZΓ
2z2−z+ 1
(z−1)2(z+ 1)dz,
where Γ is the positively oriented simple loop |z|= 4.
2. Compute the following integrals:
(1) Z2π
0
dθ
1 + sin2θ;
(2) Z∞
0
x2
(x2+ 1)(x2+ 4)dx;
(3) Z∞
−∞
cos x
(x2+ 1)(x4+ 1)dx;
(4) Z∞
0
sin(2x)
x(x2+ 1)2dx.
3. Show that
Z∞
0
dx
x3+ 1 =2π√3
9
by integrating 1/(z3+ 1) around the boundary of the circular sector Sρ:{z=reiθ :
0≤θ≤2π/3,0≤r≤ρ}and letting ρ→ ∞.
4. Find a M¨obius transformation that maps the unit disk |z|<1 onto the right half-plane
and takes z=−ito the origin.