MAT 461 - Test 2 Practice
1. (a) Let f(z) = 1
z(z−1) . Give the two Laurent Series centered at z= 0, one valid
0<|z|<1, one valid 1 <|z|<∞.
(b) Give the Laurent series centered at z=π
2for f(z) = s in z
(z−
π
2)3. What is the residue
of f(z) and z=π
2?
2. Let f(z) be entire. Assume that |f(z)|> m for all complex numbers z, where mis a
positive constant. Prove that f(z) is constant.
Let Cbe the circle |z|= 2, positively oriented.
3. Compute the integral
ZC
1
z2(z−3i)dz
4. Compute the integral
ZC(z−1)3e1
z−1dz
5. Compute the integral
ZC
e1
zcos(1
z)dz
6. Define log zwith the branch cut at −π < θ < π. Compute the integral
ZC
sin(log(z+ 3))
(z−4i)2dz
