LOYOLA COLLEGE (AUTONOMOUS), CHENNAI  600 034.
M.Sc. DEGREE EXAMINATION  PHYSICS
SECOND SEMESTER  APRIL 2003
PH 2803 / PH 825  MATHEMATICAL PHYSICS
28.04.2003
1.00  4.00
Max : 100 Marks
PART  A
(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 z z* representation for a circle.
02. State Liouville’s theorem.
z
03. Develop Laurent series of f ( z ) 
about z = -2.
( z  1)( z  2)
04. Write the Jacobian of the transformation w  2 z 2  i z  3  i .
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
...... in the interval
08. Obtain the orthonormalising constant for the series cos , cos
L
L
(L, L).

09. Evaluate
 4 x2
3
d x using the knowledge of Gamma function.
0
10. Generate L2 (x) and L3 (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. Determine a function which maps the indicated region of w plane on to the upper half of
the z  plane
v
y
w plane
z plane
p
T
Q
s
u
p1
Q1
S1
T1
-b
+b
-1
+1
13. Develop halfrange Fourier sine series for the function f (x) = x ; 0 < x < 2. Use the

(1) n 1
2
results to develop the series
.

12 n 1 n 2
X
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.
1

2

2A
15. (a) If f (x) =  Ak Pk ( x) obtain Parseval’s Identity  { f ( x)} d x   k
k 0
k 0 2 k  1
1
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)
2
PART  C
(412.5=50 marks)
Answer any FOUR.
16. Show that u (x, y) = Sin x Coshy + 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.
5

4
sin

0
ez d z
(b) Using suitable theorems evaluate  2
c : z  i 2.
(7+5.5)
c z 4
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)
2 2
(s  a )
19. Solve the boundary value problem
2 y
2 y
2

a
.
with Y (0, t) = 0; yx (L, t) = 0
t2
 x2
y (x, 0) = f (x) ; yt (x, 0) = 0 and y ( x, t )  M . and Interpret physically.
20. Solve Bessels differential equation using Froebenius power series method.
+ + + + +