LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – PHYSICS

FG 17

FOURTH SEMESTER – APRIL 2008

PH 4502 – MATHEMATICAL PHYSICS

 

 

 

Date : 26/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART-A

Answer ALL questions                                                                     (10×2=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’.
3. Evaluate .
4. What is a single valued function in a complex region.
5. Find ‘c’ if is solution to the equation

1. Write down a homogeneous first order partial differential equation.
2. Define the Fourier sine transform of a function f(x).
3. If is the Fourier transform of f(x), what is the Fourier transform of .
4. Define the shift operator on f(x) by ‘h’.
5. Write down the Simpson’s 1/3 rule for integration.

 

 

PART-B

Answer any FOUR questions                                                          (4×7¹/₂=30 marks)

 

1. Determine the roots of  and  and locate it in the complex plane.
2. If ‘C’ is a line segment from -1-i to 1+i, evaluate .
3. Derive the partial differential equation satisfied by a vibrating elastic string subject to a     tension ‘T’.
4. Obtain the Lagrange’s interpolation formula for following table:

1. Find the Fourier sine transform of exp(-at).

 

 

PART-C

Answer any FOUR questions                                                          (4×12¹/₂=50 marks)

 

1. a) Derive the Cauchy Riemann equation for a function to be analytic.          (5m)

1. b) Show that the function is harmonic and hence

construct the corresponding analytic function.                                               (7¹/₂m)

 

 

 

1. a) State and prove Cauchy’s integral theorem.                                               (5m)

1. b) Verify the Cauchy’s integral theorem for the integral of taken over the boundary of the rectangle with vertices -1, 1, 1+i and -1 +i in the counter clockwise sense. (7¹/₂m)

1. Solve the heat equation , subject to the conditions u(x=0,t)=0 and u(x=L,t)=0       for all ‘t’.

 

1. a) State and prove the convolution theorem for Fourier Transforms.             (2+3=5m)

1. b) Find the Fourier transform of the function f(x) defined in the interval –L to +L, as

(7¹/₂m)

 

1. 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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