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 z1 = a – i and z2 = a + i fine z1* z2, 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 z0 = -1-i and z= 1+i.
b).
for f(z) = 3z and g(z) = -3, and any real constants c1 and c2.
- 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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