Math 113 (Spring 2009) Yum-Tong Siu 1
Homework Assigned on February 19, 2009
due February 24, 2009
(numbering of problems continued from
the last assignment with the same due date)
Problem 4 (from Stein & Shakarchi, p.103, #2). Evaluate the integral
Z∞
−∞
dx
1 + x4.
Where are the poles of 1
1+z4?
Problem 5 (from Stein & Shakarchi, p.103, #4). Show that
Z∞
−∞
xsin x
x2+a2dx =πe−a
for all a > 0.
Problem 6 (from Stein & Shakarchi, p.103, #5). Use contour integration to
show that Z∞
−∞
e−2πixξ
(1 + x2)2dx =π
2(1 + 2π|ξ|)e−2πξ
for all ξreal.