Complex Analysis-Ii - BU 2024 Question Paper

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OP-486
M.Sc. (REG/PVT/ATKT)
Examination, 2024
(Second Semester)
MATHEMATICS
Paper-IV
Complex Analysis-II

(i)

Every function which is mesomorphic in the whole complex plane is the quotient of :

(a) Two entire functions

(b) Three entire functions

(d) None of the above

Time : 3 Hours]

[Maximum Marks :

Reg. : 85
Pvt. : 100

Note : Attempt the questions of all Sections as directed.

Section A

(Objective Type Questions) 2x5=10

(ii)

The unit circle |z| = 1 is a natural boundary of the function :

(a)

f(z) = ∑_n=0^∞ zⁿ

(b)

f(z) = ∑_n=0^∞ zⁿ

(c)

f(z) = ∑_n=0^∞ zⁿ

(d)

f(z) = ∑_n=0^∞ zⁿ ln

Section B

(Short Answer Type Questions) 5x5=25

Note : Attempt all questions. All questions carry equal marks.

(iii)

If f : G → C is an analytic function then u = Re f and v = Im f are called :

(a) Harmonic

(b) Complex

(d) Harmonic Conjugates

(iv)

Order of cos √z is :

(a) 1

(b) 1/2

(d) 0

(v)

If G is an open connected set in C and f : G → C is a continuous function, then f is called a branch of the logarithm if :

(a) z = exp f (z), ∀z ∈ G

(b) z = f (z)

(d) z = log f (z)

For any z ∈ C prove that |½z| ≤ log(1+z) ≤ &frac32;z.

Define Euler's Gamma function and prove that :

√z+1 = z√z.

Prove that there cannot be more than one analytic continuation of a function f (z) into the same domain.

Explain power series method of analytic continuation.

Prove that the Poisson Kernel Pr (θ) can be expressed as :

Pr (θ) = ^1–r²⁄_{1–2rcosθ+r²} = Re &left( ^{1+re^iθ}⁄_1–re^iθ &right)

Section C

(Long Answer Type Questions) 10x5=50

Note : Attempt all questions. All questions carry equal marks.

Define Green's function and prove that, if G be a bounded Dirichlet Region, then for each a ∈ G there is a Green's function on G with singularity at a.

Find the order of polynomial :

P(z) = a₀ + a₁z + a₂z² + .....+a_nzⁿ, a_n ≠ 0.

Let f be an entire function of finite order, then f assumes each complex number with one possible exception.

Let f be an analytic function in the disc B(α, r) such that |f'(z)–f'(α)| < |f'(α)|, for all z is B (α, r). z ≠ α then f is one-one.

State and prove the little Picard's theorem.

For Re z > 1. prove that :

G(z) = √z = ∫₀^z (t^z-1)(1-t)^-1 dt.

If |z| ≤ 1 and p ≥ 0. then prove that :

1–E_p (z) ≤ |z|^p+1, where E_p (z) is an elementary factor.

Show that the circle of convergence of power series

f(z) = ∑_n=0^∞ zⁿ is a natural boundary.

If the radius of convergence of the power series :

f(z) = ∑_n=0^∞ zⁿ

is non-zero finite, then prove that f (z) has at least one singularity on the circle of convergence.

State and prove Harnack's inequality.

Let G be a region and let a ∈ ∂∞G ⊆ G such that is a barrier for G at a, if f : ∂∞G → R is continuous and u is the Perron function associated with f then :

lim_z→a u(z) = f(a).

10.

State and prove the Jensen's formula.

State and prove the Hadamard's three circles theorem.

11.

Let f be analytic in D = { z : |z| < 1 } and let f (0) = 0, f'(0)=1 and |f (z)| ≤ M for all z in D. Then

M ≥ 1 and f(D) &supb; B &left( 0, ¹⁄_6M &right).

State and prove the Great-Picard theorem.