LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – PHYSICS
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SIXTH SEMESTER – April 2009
PH 6604 – MATHEMATICAL PHYSICS
Date & Time: 21/04/2009 / 9:00 – 12:00 Dept. No. Max. : 100 Marks
PART-A (10 X 2 =20 MARKS)
ANSWER ALL QUESTIONS
1). Write down the triangle inequality for two complex numbers z1 and z2.
2). Write down the complex representation for a circle of radius 2 units.
3). State Cauchy Riemann conditions for a function f(x,y) = u(x,y) + i v(x,y) to be analytic.
4). Define simply connected and multiply connected domains in a complex plane.
5). If u1 and u2 are two solutions of a homogeneous differential equation what can you say about
u = a u1 + b u2, with `a’ and ‘b’ constants.
6). Write down the two dimensional wave equation for a wave with velocity 1 m/s.
7). In the expansion for f(x) = a0 + n cos (nx) + n sin (nix) , write a0 in terms of f(x).
8). If f(x) is an even function of period 2, what happens to the Fourier sine coefficients.
9). Write down the Trapezoidal rule for integration of a function f(x) between x0 and x0 +h.
10.Write down the relationship between the shift operator `E’ and the forward difference
operator.
PART-B (4 X 7.5 = 3O MARKS)
ANSWER ANY FOUR QUESTIONS.
11). If f(z) = 3z2 + z, evaluate f(z) for a). z = 2 + i and z = -4 + 2 i and locate these points in the
complex plane.
12). State and prove Cauchy’s integral theorem.
13). Obtain the Laplace equation in two dimensions in terms of the polar coordinates.
14). If F(s) is the Fourier transform of f(x) find the Fourier transform of f(ax) and f(x-u), with
`a’ and `u’ being constants.
15). Using Euler method, solve the following differential equation to find y(0.4), given ,
with y(0)=1 and h = 0.1. Compare your result with the exact solution.
PART-C (4 x 12.5 = 50 MARKS)
ANSWER ANY FOUR QUESTIONS.
- What do you mean by conjugate harmonic functions? If the following functions are
harmonic, find their conjugate functions, f(x,y) = u(x,y) + i v(x,y),
a). u (x,y) = e x cos (y) ; b). v = xy.
17). Evaluate the following integrals over the unit circle.
- a) and b).
18). (i). Find `a’ and ‘b’ if u(x,y) = a x2 – b y2 is solution of the Laplace equation in two
dimensions.
(ii). Derive the partial differential equation for small transverse displacement `u’ of an
elastic string.
19). (i). State and prove Parseval’s identities for Fourier transforms.
(ii). Find the Fourier transform of f(x) = 2 for –a < x < a and f(x) = 0 for all other values.
20). From the following census data find the population for the year 1895 and 1906
| Year | 1891 | 1901 | 1911 | 1921 | 1931 |
| Population
(in thousands) |
46 | 66 | 81 | 93 | 101 |
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