LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION - PHYSICS
SECOND SEMESTER – APRIL 2008
FG 31
PH 2900 / 2803 - MATHEMATICAL PHYSICS
Date : 26/04/2008
Time : 1:00 - 4:00
Dept. No.
Max. : 100 Marks
PART – A
Answer ALL questions.
(10 x 2 = 20 marks)
1. State Cauchy’s integral formula for derivative.
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2. Check whether f ( z )  is analytic or not.
z
3. Find the Taylor expansion for a function

f ( z )  Sin z at z 
4
4. State Laurent’s Theorem.
5. Give two examples of periodic functions.
Is f ( z )  tan z periodic or not?
 1  a    a

6. Find the Fourier transform of f ( x)  
0
otherwise

7. Solve y ' ' m y  0 , where m is a constant.
8. Are the set of functions H n ( x), H m ( x) are orthogonal in the limit   to   . Expalin why?
9. Show that nLn1 ( x)  nL/ n1 ( x)  nL/ n ( x) .
10. Using the knowledge of Gamma & Beta functions evaluate,

2

tan 
d
0
PART – B
Answer any FOUR questions.
(4 x 7.5 = 30 marks)
11. Derive Cauchy-Riemann equation in polar Co-ordinate System.
12. a) Using Cauchy’s Integral formula,
ez
dz .
evaluate 
z  0.3
z 1
b) If f (z ) is analytic in a closed region, show that
 f ( z )dz  0 .
C
0 4
x
13. Expand f ( x)  
in a series of sines with a period of 8.
48
8  x
14. Show that the eigen functions belonging to two different eigen values are orthogonal with respect
to R(x) in (a, b)
1

15. Prove that  e
0
2
2
1   4
Cos d 
e
2 
PART – C
Answer any FOUR questions.
(4 x 12.5 = 50 marks)
16. Derive Poisson’s Integral formula for circle.
17. Using Cauchy’s Residue theorem,
2
e i
evaluate 
d
4

3
Cos

0
18. a) Find the Fourier coefficients corresponding to the function
5  x  0
0
f ( x)  
with the Period 10.
0 x5
3
1 1
1
2



..........

12 2 2 32
6
19. Solve the boundary value problem,
2
2 y
2  y

a
.
t 2
x 2
y(0, t )  0; y x ( L, t )  0; y( x,0)  f ( x)
b) Prove that
yt ( x,0)  0; y( x, t )  m.
20. Using the method of Frobenius, solve Laugerre’s differential equation.
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