Complex Analysis-Ii - BU 2022 Question Paper

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GH-486

M.A./M.Sc. IInd Semester (Reg./Pvt./ATKT)

Examination, 2022

Maths

Paper - IV

Complex Analysis-II

Time : 3 Hours]

[Maximum Marks :

Reg. 85

Pvt.=100

Note : - Attempt all questions.

SECTION - 'A'

7x5=35

Short Answer Type Questions

If |z| ≤ 1 and p ≥ 0, then show that |1 - E_p (z)| ≤ |z|^p+1.

Define Riemann's Zeta function and explain its extension.

Let V and U be open subsets of C with V ⊆ U and ∂V ∩ U = φ. If H is a component of U and H ∩ V = φ then H ⊆ V.

By using Mittag-Leffiers theorem to show that

π² 1
—— = Σ ———
sin² πz n=-∞ ∞ (z-n)²

Let (Γ, D) be a function element which admits unrestricted continuation in the simply connected region G. Then there is an analytic functions F : G → C such that F(z) = f(z) for all z in D.

Prove that

1 π
—— ∫ P_r(θ) dθ = 1
2π -π

If G is a bounded Dirichlet region then for each a in G, there is a Green's function on G with singularity at a.

State and prove Jensen's formula.

Let f be a analytic function on the disk B (a, r) such that |f'(z) - f'(a)| < |f'(a)| for all z ∈ B(a, r). z = a, then show that f is one-one.

State and prove Block's theorem.

SECTION - 'B'

10x5=50

Long Answer Type Questions

State and prove Great Picard's theorem.

Represent sin πz in the form of canonical product.

State and prove Monodromy theorem.

If f(z) is an entire function of order ρ and convergence expo-nent σ then show that σ ≤ ρ.

State and prove Schottky's theorem.

State and prove Borel's theorem.

State and prove Weierstrass factorization theorem.

State and prove that Rung's theorem.

State and prove Schwartz reflection theorem.

State and prove that Hadamard's factorization theorem.