LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc., DEGREE EXAMINATION – PHYSICS
FOURTH SEMESTER – SUPPLEMENTARY – JUNE 2012
PH 4504/4502/6604 – MATHEMATICAL PHYSICS
Date : 27-06-2012
Time : 2:00 - 5:00
Dept. No.
Max. : 100 Marks
SECTION – A
Answer ALL the Questions:
(10 x 2 = 20 Marks)
1. Define Analytic function.
2. Write down Laplace equation.
3. State Cauchy’s integral theorem.
4. Write down two basic properties of complex line integral.
5. Write down the steady-state two-dimensional heat flow equation.
6. State the superposition or linearity principle.
7. Define Fourier size transform.
8. State Convolution theorem.
9. Give Newton’s forward interpolation formula.
10. Write down the Trapezoidal rule.
SECTION – B
Answer any FOUR Questions:
(4 x 7.5 = 30 Marks)
11. Using the Cauchy-Riemann equations, show that f(z)=z3 is analytic in the entire z-plane.
12. Find the value of ∫c (z2+1)dz/(z2-1), if c is the circle of unit radius with centre at z=1.
13. Derive the D’ Alembert’s solution of the wave equation.
14. Discuss the properties of Fourier transforms.
15. Evaluate∫01 dx/(1+x2), using Trapezoidal rule with h=0.2.
SECTION – C
Answer any FOUR Questions:
(4 x 12.5 = 50 Marks)
16. a) Derive the Cauchy – Riemann equations.
(6.5)
b) Show that (i) cos z = cos x coshy –I sinx sin hy (ii) sin z = sin x coshy + I cos x sin hy.
(6)
17. State and prove Cauchy’s integral formula.
18. Solve the 1 – d heat flow equation by the method of separation of variables.
19. (a) Define Fourier transform of a function f(x) and find the second derivative of the Fourier transform. (6.5)
(b) Find the Fourier size and cosine transform of f(x0 = e-x.
20. Compute the value of the definite integral ∫45.2 loge x dx using (i) Trapezoidal rule (ii) Simpson’s rule.
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(6)