LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – PHYSICS
FOURTH SEMESTER – APRIL 2008
PH 4502 - MATHEMATICAL PHYSICS
Date : 26/04/2008
Time : 9:00 - 12:00
Dept. No.
FG 17
Max. : 100 Marks
PART-A
Answer ALL questions
(10x2=20 marks)
1. What is the principal value of the complex number z=1+i?
2. Write down the equation of the circle in the complex plane centered at ‘a’ with radius ‘r’.
i
3. Evaluate  (3  2 z )dz .
0
4. What is a single valued function in a complex region.
5. Find ‘c’ if u ( x, t )  x 2  at 2 is solution to the equation
 2u 2  2u
c
0
t 2
x 2
6. Write down a homogeneous first order partial differential equation.
7. Define the Fourier sine transform of a function f(x).
8. If g ( ) is the Fourier transform of f(x), what is the Fourier transform of
df
.
dx
9. Define the shift operator on f(x) by ‘h’.
10. Write down the Simpson’s 1/3 rule for integration.
PART-B
(4x71/2=30 marks)
Answer any FOUR questions
11. Determine the roots of 1  i and 1  i and locate it in the complex plane.
12. If ‘C’ is a line segment from -1-i to 1+i, evaluate  (az  b)dz .
c
13. Derive the partial differential equation satisfied by a vibrating elastic string subject to a
tension ‘T’.
14. Obtain the Lagrange’s interpolation formula for following table:
( x, y ) : (0, 0), (1,3), (2,9)
15. Find the Fourier sine transform of exp(-at).
PART-C
Answer any FOUR questions
(4x121/2=50 marks)
16. a) Derive the Cauchy Riemann equation for a function to be analytic.
(5m)
3
2
2
2
b) Show that the function u ( x, y)  x  3xy  3x  3 y  1 is harmonic and hence
construct the corresponding analytic function.
(71/2m)
1
17. a) State and prove Cauchy’s integral theorem.
(5m)
2
b) Verify the Cauchy’s integral theorem for the integral of z taken over the boundary of the
rectangle with vertices -1, 1, 1+i and -1 +i in the counter clockwise sense.
(71/2m)
u
 2u
18. Solve the heat equation
 c 2 2 , subject to the conditions u(x=0,t)=0 and u(x=L,t)=0
t
x
for all ‘t’.
19. a) State and prove the convolution theorem for Fourier Transforms.
(2+3=5m)
b) Find the Fourier transform of the function f(x) defined in the interval –L to +L, as
f ( x)  1 for  L  x   L
(71/2m)
f ( x)  0 for x   L and x   L
20. Given the following population data, use Newton’s interpolation formula to find the population
for the years 1915 and 1929
(Year, Population (in Thousands)): (1911, 12) (1921, 15) (1931, 20), (1941, 28).
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