Math 113 (Spring 2009) Yum-Tong Siu 1
Homework Assigned on February 26, 2009
due March 3, 2009
(numbering of problems continued from
the last assignment with the same due date)
Problem 6. Verify that
I|z|=4
z15
(z2+ 1)2(z4+ 2)3dz = 2πi
by using the change of variables z=1
w.
Problem 7. Evaluate the integral
Z∞
0
x1
3
1 + x2dx
by applying the method of residues to a branch of the function
f(z) = z1
3
1 + z2
defined on C−[0,∞).
Problem 8. Evaluate the integral
Z∞
0
log (1 + x2)dx
x1+α(0 < α < 1)
by applying the method of residues to a branch of a holomorphic function.
(Hint: Try integration by parts.)