# Evaluate the following integrals using Cauchy’s integral formula

August 30, 2017

MATH 132
HOMEWORK #5
DUE NOVEMBER 6 (FRIDAY)

1. Let −∞ < t < ∞. Show that

1
2
2

e−x /2 e−itx dx = e−t /2
2π −∞
2

by integrating e−z /2 around the rectangle with vertices ±R, ±R + it, and
letting R → ∞.

2

(Hint: Use the known value of the Gaussian integral : −∞ e−x = π.)

2. Evaluate the following integrals using Cauchy’s integral formula:
zn
(a) |z|=2 z−4 dz, (n ≥ 0)
(b)
(c)
(d)

zn
|z|=4 z−2 dz, (n
sin z
|z|=2 2z+π dz
ez
|z|=1 z(z 2 −4) dz

≥ 0)

3. A function f (z) on the complex plane is called doubly periodic if there
exist R-linearly independent ω1 , ω2 ∈ C such that
f (z + ω1 ) = f (z + ω2 ) = f (z),
for all z ∈ C. Show that if f (z) is entire and doubly periodic, then f (z) is
constant.
(Hint: Use the Maximum Principle: any continuous real-valued function on
a compact1 subset K ⊆ C attains its maximum and minimum values in K.)

1i.e., closed and bounded

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