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(b) 0
(c) [ei(n+1)]02π
(d) Does not exist
Total No. of Questions : 15
[Total No. of Printed Pages : 07
QR-444
M.Sc. Examination, 2024
(First Semester)
MATHEMATICS
(Compulsory)
Paper-IV
(Complex Analysis-I)
Time : 3 Hours
[Maximum Marks : 85
Note : The question paper is divided into three Sections.
Section A
5×3=15
(Objective Type Questions)
Note : Attempt all questions. All questions carry equal marks.
If n is any integer positive or negatie other than-1, then the value of ∮|z|=r Zn dz is equal to :
(a) Zn+1 / (n+1)
The function ω = 1 / sin(1/z) has an essential singularity at the :
(a) Origin
(b) All points except origin
(c) Infinity
(d) None of the point
A singularity of f (z) which has no unique limiting value is often called :
(a) a removable singularity
(b) an essential singularity
(c) Pole of order n
(d) Zero of order n
Or
The residue of z3 / ((z-1)(z-2)(z-3)) at ∞ is :
(a) 1/2
(b) -8
(c) 6
(d) -6
If the sense of rotation as well as the magnitude of the angle is preserved, the transformation is called :
(a) Isogonal
(b) Conformal
(c) Bilinear
(d) Linear
Section B
5×5=25
(Short Answer Type Questions)
Note : Attempt all questions. All questions carry equal marks.
Evaluate ∫C x2 dz, where C is an arc of the circle x = r cos θ, y = r sin θ from θ = α to θ = β.
Or
If f(z) is continuous on a contour C of length L on which it satisfies the inequality |f(z)| ≤ M, then |∫C f(z)dz| ≤ ML.
If |f(z)| has a maximum M on |z-a| = r < ρ, then |an| ≤ M/rn where an = f(n)(a)/n!
Or
Expand log(1 + z) in a Taylor's series about z = 0.
Define poles and show that poles are isolated.
Or
Find Laurent series about the indicated singularities for the following function and give the name of the singularities : (z-3)sin(1/(z+2)) = z-2.
Prove that ∫0∞ dx/(1+x2)2 = π/2.
Or
State and prove Jordan's inequality.
Or
Define bilinear and conformal transformations with example.
Section C
9×5=45
(Long Answer Type Questions)
Note : Attempt all questions. All questions carry equal marks.
If f(z) is regular in a domain D, then show that its derivative is given by f'(ξ) = 1/(2πi) ∫C f(z)/(z-ξ)2 dz, where C is any simple closed contour in D surrounding the point z=ξ.
Or
State and prove Taylor's theorem.
State and prove Liouville's theorem.
Or
State and prove Cauchy's residue theorem.
If f(z) is analytic inside and on a simple closed curve C, then the maximum value of |f(z)| occurs on C, unless f(z) is a constant. Prove that if f(z) ≠ 0 inside C, then |f(z)| must assume its minimum value on C.
Or
State and prove Rouche's theorem.
Show that ∫02π ezt/(z2+2z+2) dz = (t-1)/2 + e-t around the circle C with equation |z|=3.
Or
Use the residue theorem to show that :
∫-∞∞ cos(x)/(x2+a2)(x2+b2) dx = π/(a2-b2) (e-b/b - e-a/a)
(a > b > 0) and ∫-∞∞ sin(x)/(x2+a2)(x2+b2) dx = 0.
Or
In the transformation z = sin ω, show that the mapping of the strip -π/2 ≤ u ≤ π/2, v ≥ 0 in the ω-plane upon the upper half of the z-plane is biuniform.
Or
State and prove Morera's theorem.