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)

- Given z
_{1}= a – i and z_{2 }= a + i fine z_{1}^{* }z_{2, }for any real ‘a’. - Verify that f(z) = z is analytic.
- State two conditions for a function to be Fourier transformed.
- Define the eigen value problem for the operator
- Express the Laplacian in polar coordinates.
- State Cauchy’s integral theorem.
- Evaluate , ‘c’ is circle of radius 1.
- State Parseval’s theorem.
- Write down the difference operator and the shift operator.
- Write down trapezoidal rule for integration.

__PART-B__

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

- 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.

- 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.

- Find DAlembert’s solution of the wave equation for a vibrating string.
- 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’.

- 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)

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

a). for f(z) = z, with z_{0} = -1-i and z= 1+i.

b).

for f(z) = 3z and g(z) = -3, and any real constants c_{1} and c_{2.}

- 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.

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

for integration.

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