# Loyola College B.Sc. Physics April 2011 Mathematical Physics Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – PHYSICS

FOURTH SEMESTER – APRIL 2011

# PH 4504/PH 4502/PH 6604 – MATHEMATICAL PHYSICS

Date : 07-04-2011              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

PART-A

Answer ALL questions.                                                                                             (10 x 2 = 20 marks)

1. Given z1 = a – i and z2 = a + i fine z1* z2, for any real ‘a’.
2. Verify that f(z) = z is analytic.
3. State two conditions for a function to be Fourier transformed.
4. Define the eigen value problem for the operator
5. Express the Laplacian in polar coordinates.
6. State Cauchy’s integral theorem.
7. Evaluate , ‘c’ is circle of radius 1.
8. State Parseval’s theorem.
9. Write down the difference operator and the shift operator.
10. Write down trapezoidal rule for integration.

PART-B

Answer any FOUR questions.                                                                                   (4 x 7.5  = 30 marks)

1. a). Show that |z|2 = 1 describes a circle centered at the origin with radius 1.

b). Simplify (1+i)(2+i) and locate it in the complex plane.

1. Verify the Cauchy’s integral theorem for along the boundary of a rectangle with vertices

(0,0) , (1,0), (1,1) and (0,1) in counter clock sense.

1. Find DAlembert’s solution of the wave equation for a vibrating string.
2. If f(s) is the Fourier transform of f(x), show that F{f(ax)} = (1/a)F(s/a) and

F{f’(x)} = is F(s). Here the prime denotes differentiation with respect to ‘x’.

1. Use Euler’s method to solve, given y(0) = 1, find y(0.04) with h = 0.01.

PART-C

Answer any FOUR questions.                                                                                   (4 x 12.5 = 50 marks)

1. a) Establish that the real and complex part of an analytic function satisfies the Laplace equation.
2. b) Prove that is harmonic and find its conjugate function.                                             (6+6.5)
3. Verify

a). for f(z) = z, with z0 = -1-i and z= 1+i.

b).

for f(z) = 3z and g(z) = -3,  and any real constants c1 and c2.

1. Using the method of separation of variables obtain the solution for one dimensional

heat equation. , with u(l,t) = 0 and u(0,t)=0.

1. a) State and prove convolution theorem for Fourier transforms.
2. b) Find the Fourier sine transform of .
3. Derive the Newton’s forward interpolation formula and deduce the Trapezoidal and Simpson’s rule

for integration.

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