Download Complex Analysis 15 - Exercises - Mathematics and more Exercises Mathematics in PDF only on Docsity!
Math 113 (Spring 2009) Yum-Tong Siu 1
Homework Assigned on March 3, 2009 due March 10, 2009
Problem 1. Use the branch cut [0, 1] and the theory of residues to establish the following formula of integration.
∫ (^1)
0
√ (^4) x(1 − x) 3
(1 + x)^3
dx =
3 π 4
Hint: Follow the techniques used for the in-class computation of
∫ (^) ∞
0
xa−^1 dx 1 + x
π sin πa
(0 < a < 1)
and the notes posted on the class website for the computation of
∫ (^1)
0
dx xα(1 − x)^1 −α^
π sin απ
(0 < α < 1).
Problem 2. Let CN be the square in C with corners at the four points
( N +
(± 1 ± i).
Let a be a positive number. Evaluate the following two series
∑^ ∞
n=
n^4 + a^4
∑^ ∞
n=
n^2 n^4 + a^4
by using respectively
f (z) =
z^4 + a^4
, f (z) =
z^2 z^4 + a^4
and evaluating
n=1 f^ (n) by considering the contour integral ∫
CN
πf (z) cot πz dz
as N → ∞.
Math 113 (Spring 2009) Yum-Tong Siu 2
Problem 3 (from Stein & Shakarchi, p.105, #15). Use the Cauchy inequali- ties or the maximum modulus principle to solve the following problems:
(a) Prove that if f is an entire function that satisfies
sup |z|=R
|f (z)| ≤ ARk^ + B
for all R > 0, and for some integers k ≥ 0 and some constants A, B > 0, then f is a polynomial of degree ≤ k.
(b) Show that if f is holomorphic in the unit disk, is bounded, and converges uniformly to zero in the sector θ < arg z < ϕ as |z| → 1, then f = 0.
(c) Let w 1 , · · · , wn be points on the unit circle in the complex plane. Prove that there exists a point z on the unit circle such that the product of the distances from z to the points wj , 1 ≤ j ≤ n, is at least 1. Conclude that there exists a point w on the unit circle such that the product of the distances from w to the points wj , 1 ≤ j ≤ n, is exactly equal to 1.
(d) Show that if the real part of an entire function f is bounded, then f is constant.
