# LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

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SUPPLEMENTARY SEMESTER EXAMINATION – JUN 2006
M.Sc. DEGREE EXAMINATION
PH 2803/PH 2900 - MATHEMATICAL PHYSICS
Date &amp; Time : 27/062006/9.00 - 12.00
Dept. No.
PART  A
Max. : 100 Marks
(10 2=20 marks)
Answer ALL questions.
01. Starting from the general equation of a circle in the xy plane, A(x2 +y2) + Bx + Cy +D=0
arrive at the zz* representation for a circle.
02. State Cauchy’s integral formula for derivatives
z
03. Develop Taylor’s series of f ( z ) 
about z = -1.
( z  1)( z  2)
04. Express z  i / z  i in the form of a+ib
1
05. Show that the Dirac delta function  (ax)   ( x) for a  0 .
a
06. State convolution theorem.
07. Solve the differential equation T  ’ + k 2 T  0 .
x
2x
, sin
...... in the interval
08. Obtain the orthonormalising constant for the series sin
L
L
(L, L).

09. Evaluate
 4 x2
3
d x using the knowledge of Gamma function.
0
10. Generate L3 (x) and L4(x) using Rodrigue’s formula for Laugerre polynomials.
PART  B
(4 7.5=30 marks)
Answer any FOUR.
11. Obtain Cauchy Rieman equations from first principles of calculus of complex numbers.
12. State and prove Cauchy’s residue theorem
13. Develop halfrange Fourier sine series for the function f (x) = x ; 0 &lt; x &lt; 2. Use the

(1) n 1
2
results to develop the series
.

12 n 1 n 2
14. Verify that the system y11 +  y  0 ; y1(0) = 0 and y (1) = 0 is a SturmLiouville System.
Find the eigen values and eigen functions of the system and hence form a orthnormal set
of functions.
15. (a) If f (x) =

1
k 0
1
 Ak Pk ( x) obtain Parseval’s Identity

2
2 Ak
k 0 2 k  1
2
 { f ( x)} d x  
where Pk (x) stands for Legendre polynomials.
/
(b) Prove that H n (x) = 2n - 1 Hn (x) where Hn (x) stands for Hermite
polynomials.(4+3.5)
PART  C
(412.5=50 marks)
Answer any FOUR.
16. Show that u (x, y) = Sin x Cosh y + 2 Cos x Sinhy + x2 +4 xy  y2 is harmonic Construct
f (z) such that u + iv is analytic.
2
d
17. (a) Evaluate 
using contour integration.
4  3 sin 
0
ez d z
(7+5.5)
c z 2  16 c : z  i  5 .
18. (a) The current i and the charge q in a series circuit containing an inductance L and
dq
di q
  E and i =
capacitance c and emf E satisfy the equations L
. Using
dt c
dt
Laplace Transforms solve the equation and express i interms of circuit parameters.
1
(b) Find L1 2
, where L-1 stands for inverse Laplace transform.
(3.5)
(s  a 2 ) 2
(b) Using suitable theorems evaluate
19. Solve the boundary value problem
2 y
2 y
2
.
with Y (0, t) = 0; yx (L, t) = 0

a
t2
 x2
y (x, 0) = f (x) ; yt (x, 0) = 0 and y ( x, t )  M .
20. Solve Bessels differential equation using Froebenius power series method.
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