Department of Mathematics
Christopher Newport University
Math 355-01 Complex Variables Spring Term 2007
Homework # 10
Due Wednesday April 11, 2007
1. Find the residue at z= 0 of the function
(a) z+ 2
z2+z
(b) exp(z)
z3(1 + z)
2. In each case, show that any singular point of the function is a pole. Determine the order mof each pole, and find the
corresponding residue.
(a) 2z+ 3
z3+z2
(b) coshz
z2+π2
3. Show that
(a) Resz=−i
Logz
(z2+ 1)2=π−2i
8
(b) Resz=−i
z1/2
(z2+ 1)2=1 + i
8√2(|z|>0,−π < arg(z)< π )
4. Find the integral
ZC
4z2+ 1
(z+ 1)2(z2+ 1) dz
taken counterclockwise around the circle (a)|z|= 2 ; (b)|z+ 1|= 1.