LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – PHYSICS
SIXTH SEMESTER – APRIL 2008
PH 6604 / 6601 - MATHEMATICAL PHYSICS
Date : 21/04/2008
Time : 9:00 - 12:00
Dept. No.
FG 24
Max. : 100 Marks
PART – A
Answer ALL the questions:
(10 x 2 = 20)
i
1. Plot z  re for a fixed ‘r’ and 0     .
1
2i
2. Find a) Re
b) Im
.
2i
3  4i
3. Give the condition on a function f(z)=u(x,y) + i v(x,y) to be analytic.
4. Evaluate  (4 z  3)dz along a straight line from i to 1+i.
5. Write down the Lplace’s equation in two dimensions in polar coordinates.
6. Write down the equation for one dimensional heat flow.
7. State Parsavel’s theorem.
8. Define Fourier cosine transform.
9. Give trapezoidal formula for integration.
10. Define the forward and backward difference operators.
PART – B
Answer any FOUR questions.
(4 x 7\ = 30)
2
2
11. Show that the following function u ( x, y )  x  y is harmonic and hence find the
corresponding analytic function, f ( z )  u ( x, y )  iv( x, y ) .
12. Prove Cauchy’s integral theorem.
13. Find D’ Alembert’s solution of the vibrating string.
14. State and prove convolution theorem in Fourier transform. (2+5\=7\)
1
15. Use Simpson’s 1/3 rule to find
 x dx correct to two decimal places, taking step size h=0.25.
0
PART – C
Answer any FOUR questions.
16. a) Determine 1  i and plot its graph
(14 x 12\ = 50)
(3\)
2
2
 (1  i) 
 (1  i) 
2
b) Perform the following operations i) (3  2i)(2  i) ii) 3 

 and locate
 (1  i) 
 (1  i) 
these values in the complex plane. (4+5=9)
17. a) State and prove Cauchy’s integral formula.
z2 1
b) Integrate 2
in the counter clockwise sense around a circle of radius 1 with centre at z=1/2.
z 1
(2+5+5\=12\)
18. Derive the Laplace’s equation in two dimensions and obtain its solutions.
19. Find the Fourier transform and the Fourier cosine transform of the function
 ( x)  e
 x2
2
.
20. Estimate the value of f(22) and f(42) from the following data by Newton’s interpolation:
x
20
25
30
35
40
45
f(x)
354
332
291
260
231
204
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1