Homework 1 – Math 440/508
Due Friday September 19 at the end of lecture
Problem 1. Show the following:
a. If Im(a) > 0, Im(b) < 0 and b 6= a,
Z
Im(a) ∞ log |x − b|
dx = log |b − a|.
2
π
−∞ |x − a|
b.
Z
∞
π
sin(x2 )
dx = .
x
4
∞
cos x
√ dx =
x
0
c.
Z
0
d.
∞
Z
0
e.
r
π
.
2
2π cos π3 a + π6
xa−1
√
dx
=
, 0 < a < 2.
1 + x + x2
3 sin(πa)
∞
(log x)2
16 π 3
√ .
dx
=
1 + x + x2
81 3
0
R∞
Hint: Try computing 0 (1 + x + x2 )−1 log x dx first.
Z
f.
Z
2π
2
Z
π
log sin θ dθ = −4π log 2.
log sin (2θ) dθ = 4
0
0
Hint: Take the function f (z) = log(1 − e2iz ) as the integrand and use a
suitable contour.
g.
∞
sin(ax)
1 + e−a
1
dx
=
−
2πx
−a
e
−1
4(1 − e ) 2a
0
Hint: Use a rectangular contour, conveniently indented.
Z
Problem 2. Let f be an entire function and let a, b ∈ C be such that |a| < R,
|b| < R. Evaluate the following integral:
Z
f (z)
dz.
|z|=R (z − a)(z − b)
Use this result to give another proof of Liouville’s theorem (the only bounded
entire functions are the constants).
1