+3-II-S-CBCS(MS)-Sc(H)-Core-V-Phy-R&B

2024

PART-I | Answer the following questions. | 1 x 8

• x	is even/odd function.

• $\int_{-\pi}^\pi \sin nx . \cos mx \, dx= $ __.
• Value of $H_0(x)$ is __.
• x=0 is a singular point for Legendre differential equation (True/False).
• Value of $Y_{00} (\theta, \phi)$ is __.
• Value of $\Gamma (3)$ _____.
• Write Laplace’s equation.
• Wave equation in one dimension is given by __.

PART-II | Answer any eight | 1.5 x 8

• State Parseval’s theorem.
• Prove that derivative of odd function is even.
• Write the generating function for Hermite polynomials.
• Find the value of $H_2(-x) + H_2(x)$.
• Define regular singular point.
• Evaluate $P_1^1(x)$.
• Evaluate $\int_0^\infty x^{1/2}e^{-x} dx$.
• Prove that $\beta (m,n)= \beta(n,m)$.
• Write boundary conditions for a dielectric sphere placed in an external uniform electric field.
• Evaluate $\Gamma (1/2)$.

PART-III | in maximum 75 words. | 2x8

• State Dirichlet’s conditions.
• Stating various conditions for f(x), write expression of Fourier integral for it.
• Obtain sine series from Fourier series.
• Prove that $P_n’ (1)= n(n+1)/2$.
• Determine ordinary point of differential equation $xy”+(\sin x)y’ +x^2y=0$.
• Evaluate $\int_0^{\pi/2} \cos^r \theta d\theta$ for $r\gt -1$.
• Prove that $\Gamma (z+1)= z\Gamma (z)$.
• Prove that $H_n’ (x)=2nH_{n-1}(x)$.
• Solve $3 \partial u/\partial x +2 \partial u/\partial y=0$ with $u(x,0)= 4e^{-x}$.
• Solve $\partial^2 u/\partial x^2 - \partial u/\partial x=0$.

PART-IV | Answer within 500 words each. | 6x4

4) Expand $f(x)= x^2$ for $-\pi \le x\le \pi$ in a Fourier series and prove that $\sum_{n=1}^\infty 1/n^2 = \pi^2 /6$.

OR

Expand in Fourier series, $f(x) = -1 \text{ for } -\pi \le x \le 0
= +1 \text{ for } 0 \le x \le \pi $.

5) Solve Hermite differential equation using Frobenius method

OR

Prove that $\int_{-1}^1 P_m ( x) P_n (x) dx = 2\delta_{mn}/(2n+1)$ and also evaluate $\int_{-1}^1 x P_n (x) dx$ for all integral values of n.

6) Prove that $(2n+1)xP_n(x) = (n+1)P_{n+1}(x) +nP_{n-1}(x);$ $xP_n’(x) = nP_n(x) +P_{n-1}’(x)$.

OR

Define Beta and Gama functions and derive the relation among them.

7) Solve Laplace’s equation in cylindrical co-ordinates.

OR

Find electric field and potential for a conducting sphere placed in an external uniform electric field. ectric field. d prove change of scale property of Laplace Transform.

b. Using Laplace transform derivatives, find Laplace transform of $\cos kt$.

OR:

Study the Damped harmonic oscillator problem and obtain its solution using Laplace transform derivatives.pression for Interference in wedge-shaped thin films.

OR

With neat diagram describe the principle, construction, and production of interference fringes with fabry-perot interferometer.

7) Explain Fraunhoffer diffraction at a double slit. Then derive an expression for intensity distribution and find the positions of maxima and minima.

OR

Describe the Fresnel’s theory of half period zones for plane waves using Fresnel’s basic assumption. ¤¤¤¤¤