IGNOU MSCPH MPH 1 Mathematical Methods in Physics Solved Assignment ENGLISH

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MSCPH 2026 - ENGLISH

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MPH 1: Mathematical Methods in Physics

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Title Name	IGNOU MSCPH MPH 1 Mathematical Methods in Physics Solved Assignment ENGLISH
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MSCPH
Course Name	Master of Science (Physics)
Subject Code	MPH 1
Subject Name	Mathematical Methods in Physics
Year	2026
Session	-
Language	ENGLISH
Assignment Code	MPH 1/Assignment-1/2026
Product Description	Assignment of MSCPH (Master of Science (Physics)) 2026. Latest MPH 01 2026 Solved Assignment Solutions
Format	Ready-to-Print PDF (.soft copy)

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MPH 001 (January 2026 - July 2026) - ENGLISH

Tutor Marked Assignment

MATHEMATICAL METHODS IN PHYSICS

Course Code: MPH-001

Assignment Code: MPH-001/TMA/2026

Max. Marks: 100

Note: Attempt all questions. The marks for each question are indicated against it.

PART A

1. a) Reduce the following PDE into three ODEs:

$$\left[ \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right] f(x, y, z) + k^2 f(x, y, z) = 0$

b) Derive an integral equation corresponding to the ODE:

$$y'' - 2y = 0$
subject to the conditions: $y(0) = 4; y'(0) = -2$

c) Use the method of separation of variables to reduce the Laplace's equation $\nabla^2 f = 0$ into three ODEs.

d) Using the generating function for Bessel functions of the first kind and integral order

$$g(x, t) = \exp \left[ \frac{x}{2} \left( t - \frac{1}{t} \right) \right] = \sum_{n=-\infty}^{\infty} J_n(x) t^n$

Obtain the recurrence relation

$$J_{n-1}(x) + J_{n+1}(x) = 2 \frac{n}{x} J_n(x)$

Also using the generating function show that

$$J_0(x) + 2J_2(x) + 2J_4(x) + 2J_6(x) + \dots + 2J_{2k}(x) + \dots = 1$

2. a) Obtain orthogonality relation for Hermite polynomials using the generating function:

$$e^{2xt-t^2} = \sum_{n=0}^{\infty} H_n(x) \frac{t^n}{n!}$

b) i) Show that the following vectors

$$\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}$

are linearly independent.

ii) The first Pauli matrix is

$$\sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
calculate $U_1(\theta) = \exp(i\theta \sigma_1) = 1 + i\theta \sigma_1 - \frac{\theta^2}{2} \sigma_1^2 - \dots$

For real $\theta$ , show that U₁ is unitary and has determinant 1.

c) Obtain the eigenvalues and eigenvectors of matrix A:

$$A = \begin{bmatrix} 2 & 0 & -1 \\ 0 & 2 & 0 \\ -1 & 0 & 2 \end{bmatrix}$
d) Define covariant and cotravariant tensors of rank 2. Prove that $a_i = g_{ij}v^j$ transform covariantly, where g_ij are the components of the matrix tensor of rank 2 and vⁱ the components of a contravariant vector.

PART B

3. a) i) Obtain the analytic function whose real part is

$$u(x, y) = e^x \cos y$

ii) Locate and name the singularity of the function:

$$\frac{z^2 - 2z}{z^2 + 2z + 2}$
b) Calculate the value of the integral $\oint_C \frac{\cos z}{z} dz$ when C is the circle $|z| = 2$ .
c) Show that the Series $\sum_{n=1}^{\infty} z^n (1 - z)$ converges for |z| < 1 and find its sum.
d) Obtain the Laurent series expansion of $\frac{e^z}{(z - 1)^2}$ about $z = 1$ . Determine the type of singularity and the region of convergence.
e) Evaluate the value of the contour integral $6 \oint_C \frac{z \, dz}{(z - 1)(z^2 + 9)}$ where C is a circle defined by $|z| = 4$ .

4. a) Evaluate the integral

$$\int_0^{2\pi} \frac{d\theta}{1 + p \cos \theta}$

by the method of residues when -1 < p < 1.

b) Consider a triangle T in the z-plane with vertices at i, 1 - i, 1 + i. Determine the triangle T₀ into which T is mapped under the transformations

$$w = iz + 2 - i$

c) Obtain the Fourier cosine transformation of the function:

$$f(x) = e^{-px}, \quad 0 < x < \infty, \quad p > 0$

d) Define homomorphisms. When do the homomorphisms become endomorphism and isomorphism?

MPH 1 2025 - English

Tutor Marked Assignment

MATHEMATICAL METHODS IN PHYSICS

Course Code: MPH-001

Assignment Code: MPH-001/TMA/2025

Max. Marks: 100

Note: Attempt all questions. The marks for each question are indicated against it.

PART A

a) The 1-D wave equation for e.m. wave propagation in free space in given by (for $vec{E}parallelhat{y}$ )

$frac{partial^2 E_y}{partial x^2}-frac{1}{c^2}frac{partial^2 E_y}{partial t^2}=0$

Solve this equation if E_y = 0 at x = 0 and x = L.

b) The Helmholtz equation in Cartesian coordinates can be written as

$left(frac{partial^2}{partial x^2}+frac{partial^2}{partial y^2}+frac{partial^2}{partial z^2} ight)f(x,y,z)+kappa^2 f(x,y,z)=0$

Reduce it to three ODEs.

c) Using the generating function for Bessel functions of the first kind and integral order, obtain the recurrence relation

$J_{n-1}(x)+J_{n+1}(x)=2frac{n}{x}J_n(x)$

d) Using the Rodrigue's formula of Legendre polynomials, obtain the values of P₃(x) and P₄(x).

e) Write an expression for generating function for Hermite polynomials. Using this expression evaluate the integral

$int_{-infty}^{+infty}xe^{-x^2}H_n(x)H_m(x),dx$

2. a) What are orthonormal vectors? Show that two non-null, orthogonal vectors are linearly independent.

b) Show that the set of all 2x2 Hermitian matrices form a four dimensional real vector space. Obtain a suitable basis for this vector space.

c) Obtain the eigenvalues and the orthonormal eigenvectors for the following real symmetric matrix:

$egin{bmatrix}cos heta&sin heta&0\sin heta&-cos heta&0\0&0&-1end{bmatrix}$

d) Define contravariant and covariant tensors of rank 2. Prove that vⁱ = g^ijaj transform contravariantly.

PART B

3. a) Using the method of Residues, prove that

$int_{0}^{pi}frac{d heta}{1+sin^2 heta}=frac{pi}{sqrt{2}}$

b) Show that the series $sum_{n=1}^{infty}frac{z^{n-1}}{2^n}$ converges for IzI < 2 and find its sum.

c) Write the Laurent series expansion of ez/ (z-1)² singularity and the region of convergence. about z = 1. Determine the type of

d) Obtain the images of the line x = 0 and y = 0 under the transformation w = z² and prove that they intersect at right angles.

4. a) Obtain the Fourier cosine transform of the function

$f(x)=e^{-lambda x},quad 0<x<infty,quadlambda>0$

b) Solve the initial value problem using the method of Laplace transform

$y''-2y'-3y=0,quad y(0)=1,quad y'(0)=7$

c) A metal plate covering the first quadrant of the xy plane has its edge along y-axis insulated. The edge along x-axis is held at an initial temperature

$T(x,0)=egin{cases}100(2-x)& ext{for}0<x<2\0& ext{for}x>2end{cases}$

Obtain the steady state temperature distribution as a function of x and y.

d) Determine the inverse Laplace transform of

$F(x)=frac{x+1}{x^3+x^2-6x}$

5. a) Construct the multiplication table for the group of permutations of {1,2,3}.

b) Obtain all the proper subgroups of the permutation groups S₃ = {e,p1...,p5 }and their cosets.

Given:

Image ignouacademy-com-ignou-mph-1-solved-assignment-html-p-ignou-49594

c) Show that in the x - y plane, a rotation about the origin in an anti-clockwise

direction by an angle φ will move the points (x, y) to by the matrix

$egin{pmatrix}cosphi&-sinphi\sinphi&cosphiend{pmatrix}$

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Course: Master of Science (Physics) (MSCPH)
Subject: MPH 1 - Mathematical Methods in Physics
Session: January 2026 - December 2026
Language: ENGLISH

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Product Information

Course Name	Master of Science (Physics)
Course Code	MSCPH
Subject Code	MPH 1
Subject	Mathematical Methods in Physics
Programme	MASTER DEGREE PROGRAMMES Courses
Language	ENGLISH
Material Type	Solved Assignment – Soft Copy / PDF

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