LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – PHYSICS

FOURTH SEMESTER – APRIL 2012

PH 4504/4502/6604 – MATHEMATICAL PHYSICS

 

 

 

Date : 21-04-2012              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 find z₁^* z_2, for any real ‘a’.
2. Verify that f(z) = x²-y²+2ixy is analytic.
3. Evaluate.
4. Define the eigen vale problem for the operator.
5. Find ‘c’ if u(x,t) = x²+at² is a solution to the wave equation

.

6. What is singular point of a complex function in a region.
7. Write down a homogeneous first order partial differential equation in two variables.
8. State Parsavel’s theorem.
9. Write down the difference operator for f(x) by ‘h’.
10. Write down trapezoidal rule for integration.

PART-B

 

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

 

11. a). Show that |z-i|² = 1 describes a circle centered at the (0,i) with radius 1.

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

12. If ‘C’ is a line segment from -1-i to 1+i evaluate .
13. Derive the partial differential equation satisfied by a vibrating elastic string subject to

a tension ‘T`.

14. 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’.

15. Obtain the Lagrange’s interpolation polynomial of degree two for the following data:

(x,y): (0,0),(1,3),(2,9)

 

 

 

PART-C

 

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

 

16. Establish that the real and complex part of an analytic function satisfies the Laplace equation.

 

17. a) State and prove Cauchy’s integral theorem.
18. b) Verify the integral theorem for , where c is a circle of radius 1.
19. Obtain the Laplacian operator in polar form from the Cartesian form.
20. a) State and prove convolution theorem for the Fourier transforms.
21. b) Find the Fourier sine transform of .
22. Derive the Newton’s forward interpolation formula and deduce the Trapezoidal and Simpson’s rule

for integration.

 

 

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