Homework 9 03–26–08
Complex Analysis, M381D 59180
1. (∼Ahlfors 4.5.3.3) Evaluate the following integrals by the method of residues:
(a)Zπ/2
0
dx
a+ sin2x,|a|>1,
(d)Z∞
0
x2dx
(x2+a2)3, a real,
(c)Z∞
−∞
x2−x+ 2
x4+ 10x2+ 9 dx ,
(g)Z∞
0
x1/3
1 + x2dx ,
2. (Prelim Aug 1997) By integrating (e2iz −1−2iz) around a suitable contour, evaluate
Z∞
0
sin2x
x2dx .
3. (Prelim Jan 1996) Evaluate the integral
Z∞
−∞
eωx
1 + exdx ,
where ωis a complex number with 0 <Re(ω)<1.
(Hint for Problem 3. Compare with the integral along R+ 2πi.)
4. (∼Ahlfors 5.2.5.3) Using Cauchy’s formula and the fact that R∞
0e−t2dt =1
2√π,
compute the Fresnel integrals
Z∞
0
sin(x2)dx , Z∞
0
cos(x2)dx .
Answer: Both are equal to 1
2pπ/2.
5. (∼Ahlfors 5.1.3.4) Show that the Laurent development of (ez−1)−1at the origin is
of the form
1
z−1
2+
∞
X
k=1
(−1)k−1B∗
k
(2k)!z2k−1.
The numbers B∗
kare known as Bernoulli numbers †. Verify that
n
X
k=1
(−1)k−12n+ 2
2kB∗
k=n , n = 1,2,3, . . . ,
and use this identity to compute B1, . . . , B3.
6. (∼Ahlfors 5.1.3.5) Express the Laurent development of cot(z) in terms of the Bernoulli
numbers.
†These are the “original” Bernoulli numbers. They are related to the Bernoulli numbers Bkbut different.
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