Math 113 (Spring 2009) Yum-Tong Siu 1
Homework Assigned on February 24, 2009
due March 3, 2009
Problem 1 (from Stein & Shakarchi, p.104, #6). Show that
Z∞
−∞
dx
(1 + x2)n+1 =1·3·5· · · (2n−1)
2·4·6· · · (2n)·π.
Problem 2 (from Stein & Shakarchi, p.104, #7). Prove that
Z2π
0
dθ
(a+ cos θ)2=2πa
(a2−1)3
2
whenever a > 1.
Problem 3 (from Stein & Shakarchi, p.104, #10). Show that if a > 0, then
Z∞
0
log x
x2+a2dx =π
2alog a.
Hint: Use as contour of integration the boundary of the indented upper half
disk ½z=x+iy ∈C¯
¯
¯
¯
y≥0, ε ≤ |z| ≤ R¾
with 0 < ε < R < ∞.
Problem 4 (from Stein & Shakarchi, p.105, #12). Suppose uis not an integer.
Prove that ∞
X
−∞
1
(u+n)2=π2
(sin πu)2
by integrating
f(z) = πcot πz
(u+z)2
over the circle |z|=RN=N+1
2(Nan integer and |N| ≥ |u|), adding the
residues of finside the circle, and letting Ntend to infinity.
