LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION - MATHEMATICS
THIRD SEMESTER – APRIL 2012
MT 3811 - COMPLEX ANALYSIS
Date : 24-04-2012
Time : 1:00 - 4:00
Dept. No.
Max. : 100 Marks
Answer all the questions.
2𝜋 𝑒 𝑖𝑠
𝑑𝑠
𝑒 𝑖𝑠 −𝑧
1. a) Prove that ∫0
= 2𝜋 if |𝑧| < 1using Leibniz’s rule
OR
b) Let 𝑝(𝑧) be a non-constant polynomial. Prove that there is a complex number 𝑎 such that 𝑝(𝑎) = 0.
(5)
1
c) Let 𝑓: 𝐺 → ℂ be an analytic function. Prove that 𝑓(𝑧) = 2𝜋𝑖 ∫𝛾
𝑓(𝑤)
𝑑𝑤
𝑤−𝑧
for |𝑧 − 𝑎| < 𝑟 where 𝛾(𝑡) =
𝑎 + 𝑟𝑒 𝑖𝑡 . Hence prove that if f is analytic in an open disk 𝐵(𝑎; 𝑅) ⊂ 𝐺 then prove that 𝑓(𝑧) =
1
(𝑛)
𝑛
∑∞
(𝑎).
𝑛=0 𝑎𝑛 (𝑧 − 𝑎) for |𝑧 − 𝑎| < 𝑟 where 𝑎𝑛 = 𝑛! 𝑓
OR
d) State and prove homotopic version of Cauchy’s theorem.
(15)
2. a) State and prove Morera’s theorem.
OR
b) Prove that a differentiable function 𝑓 on [𝑎, 𝑏] is convex if and only if 𝑓’ is
increasing.
c) State and prove the Arzela-Ascoli theorem.
OR
d) State and prove the Riemann mapping theorem.
(5)
(15)
1
1
3. a) Show that 𝛾 = lim [(1 + 2 + ⋯ + 𝑛) − 𝑙𝑜𝑔𝑛] in the usual notation.
𝑛→∞
OR
b) If |𝑧| ≤ 1 and 𝑝 ≥ 0 then prove that |1 − 𝐸𝑝 (𝑧)| ≤ |𝑧|𝑝+1.
(5)
c) (i) Let (𝑋, 𝑑) be a compact metric space and let {𝑔𝑛 } be a sequence of continuous functions from X
into ℂ such that ∑ 𝑔𝑛 (𝑥) converges absolutely and uniformly for x in X. Then prove that the
product 𝑓(𝑥) = ∏∞
𝑛=1(1 + 𝑔𝑛 (𝑥)) converges absolutely and uniformly for x in X. Also prove that
there is an integer 𝑛0 such that 𝑓(𝑥) = 0 if and only if 𝑔𝑛 (𝑥) = −1 for some n, 1 ≤ 𝑛 ≤ 𝑛0 .
(ii) State and prove Weierstrass factorization theorem.
(7+8)
OR
d) Let 𝑅𝑒𝑧𝑛 > −1, then prove that ∑ 𝑙𝑜𝑔(1 + 𝑧𝑛 ) converges absolutely if and only if ∑ 𝑧𝑛 converges
absolutely.
e) State and prove Bohr-Mollerup theorem.
(7+8)
4. a) State and prove Jensen’s formula.
OR
𝑑(𝑠,𝑡)
b) If (𝑆, 𝑑) is a metric space, then prove that 𝜌(𝑠, 𝑡) = 1+𝑑(𝑠,𝑡) is also a metric on 𝑆.
(5)
c) State and prove Rung’s theorem.
(15)
OR
d) State and prove Hadamard’s factorization theorem.
(15)
5. a) Prove that any two bases of a same module are connected by a unimodular transformation.
OR
b) Prove that an elliptic function without poles is a constant.
(5)
c) (i) Prove that the zeros 𝑎1 , 𝑎2 , … , 𝑎𝑛 and poles 𝑏1 , 𝑏2 , … , 𝑏𝑛 of an elliptic function satisfy 𝑎1 + 𝑎2 +
⋯ + 𝑎𝑛 ≡ 𝑏1 + 𝑏2 + ⋯ + 𝑏𝑛 (𝑚𝑜𝑑 𝑀).
(ii) Derive Legendre’s relation
(7+8)
OR
d) (i) State and prove the addition theorem for the Weierstrassfunction.
(𝑧)
′ (𝑧)
1
′ (𝑢)
1| = 0
(ii) Show that | (𝑢)
′
(𝑢 + 𝑧)  (𝑢 + 𝑧) 1
(8+7)
*******************