Math 113 (Spring 2009) Yum-Tong Siu 1
Homework Assigned on February 5, 2009
due February 17, 2009
Problem 1 (from Stein & Shakarchi, pp.30-31, #25).
(a) Evaluate the integral
Zγ
zndz
for all integers n(positive, negative, or zero). Here γis any circle
centered at the origin with the positive (counterclockwise) orientation.
(b) Same question as before, but with γany circle not containing the origin.
Hint: Use the parametrization z=a+reiθ for 0 ≤θ≤2πfor the circle
of center a∈Cand radius r > 0. Find a complex-valued function F(θ)
of the real variable θfor 0 ≤θ≤2πsuch that
d
dθ F(θ) = a+reiθ nd
dθ a+reiθ
for 0 ≤θ≤2π. Consider F(2π)−F(0). Distinguish between the case
where |a|< r and the case where |a|> r.
(c) Show that if |a|< r < |b|, then
Zγ
1
(z−a)(z−b)dz =2πi
a−b,
where γdenotes the circle centered at the origin, of radius r, with the
positive orientation.
Hint: Use the decomposition of 1
(z−a)(z−b)into partial fractions A
z−a+B
z−b
(where Aand Bare complex numbers) and use Part (b).
Problem 2 (from Stein & Shakarchi, p.64, #1). Prove that
Z∞
x=0
sin x2dx =Z∞
x=0
cos x2dx =√2π
4.
These are the Fresnel integrals. Here, R∞
0is interpreted as limR→∞ RR
0.
