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Total No. of Questions : 11
[Total No. of Printed Pages : 8
KL-486
M. A./M. Sc. Mathematics (Sem. II)
Regular/Private/ATKT/EX
Examination - June - 2023
MATHEMATICS
Paper IV
Complex Analysis-II
Time : 3 Hours
Maximum Marks :
Reg. : 85
Pvt. : 100
Reg. : 85
Pvt. : 100
Note : Attempt all questions.
Section A
2×5=10
Objective Type Questions
1.
Choose the correct answer :
(i)
The Riemann zeta function is defined by
the equation \(\zeta(z) = \sum_{n=1}^{\infty} n^{-z}\) for :
(ii)
The radius of convergence of power series
\(\sum_{n=0}^{\infty} \frac{z^n}{n^{2n+1}}\) is :
(iii)
If Pr (\(\theta\)) is Poisson kernel, then the value
of \(\frac{1}{2\pi} \int_{-\pi}^{\pi} \text{Pr}(\theta)d\theta\) is :
(iv)
If \(f\) is an entire function of finite order
\(\lambda\), then \(f\) has finite genus \(\mu \le \lambda\) is known
as :
(v)
If \(f\) is an entire function that omits two
values, then \(f\) is a constant known as :
Section B
5×5=25
Short Answer Type Questions
2.
For \(z \ne 0, -1, \ldots\), prove that :
\( \left| \frac{z}{z+1} \right| \le 2\left| z \right| \)
Or
Let \(S = \{z : \operatorname{Re} \, z \ge a\}\), where \(a > 1\). If \(\epsilon > 0\),
then prove that there is a number \(\delta\), \(\delta \in (0, 1)\),
such that for all \(z\) in \(S\) :
\( \left| \int_{\alpha}^{\beta} (e^t - 1)^{-1} t^{z-1} \, dt \right| < \epsilon \)
where \(\delta < \beta > \alpha\).
3.
Write the statement of Mittag-Leffler's theorem.
Or
Prove that there cannot be more than one
analytic continuation of a function \(f(z)\) into the
same domain.
4.
Define Poisson kernel \(\text{Pr}(\theta)\) and prove that :
\( \text{Pr}(-\theta) = \text{Pr}(\theta) \)
Or
Define Green function.
5.
Define the following :
Or
Find the order of the function \(e^{az}\).
Section C
10×5=50
Long Answer Type Questions
6.
Let \(f\) be an analytic function on the disk
\(B(a; r)\) such that \(|f'(z) - f'(a)| < |f'(a)|\) for
all \(z\) in \(B(a; r)\), \(z \ne a\), then prove that \(f\) is one-
one.
Or
If \(f(z)\) is univalent in a domain \(D\), show that
\(f'(D) \ne 0\) in \(D\).
7.
If \(\operatorname{Re} \, z > 0\), then prove that :
\( \overline{\zeta(z)} = \int_{0}^{\infty} e^{-t} t^{z-1} \, dt \)
Or
If \(\operatorname{Re} \, z > 1\), then prove that :
\( \zeta(z) = \prod_{n=1}^{\infty} \left( \frac{1}{1 - p_n^{-z}} \right) \)
8.
Using Mittag-Leffler's theorem to show that :
\( \frac{\pi^2}{\sin^2 \pi z} = \sum_{n=-\infty}^{\infty} \frac{1}{(z-n)^2} \)
Or
Let \(y : [0, 1] \to C\) be a path from \(a\) to \(b\) and
let:
\( \{ (f_t, D_t) : 0 \le t \le 1 \} \) and \( \{ (g_t, B_t) : 0 \le t \le 1 \} \)
be analytic continuations along \(y\) such that
\( [f_0]_a = [g_0]_a \). Then prove that \( [f_1]_b = [g_1]_b \).
9.
State and prove Harnack's theorem.
Or
Let \(G\) be a region and let \(a \in \partial G\) such that
there is a barrier for \(G\) at \(a\). If \(f : \partial_\infty G \to R\)
is continuous and \(u\) is the Perron function
associated with \(f\), then \( \lim_{z \to a} u(z) = f(a) \).
10.
State and prove Hadamard's three circles
theorem.
11.
State and prove Great Picard Theorem.
Or
State and prove \(\frac{1}{4}\)-theorem.