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PHE 14: Mathematical Methods in Physics-III

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Title Name	IGNOU PHE 14 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSC
Course Name	Bachelor in Science
Subject Code	PHE 14
Subject Name	Mathematical Methods in Physics-III
Year	2026
Session	-
Language	English Medium
Assignment Code	PHE 14/Assignment-1/2026
Product Description	Assignment of BSC (Bachelor in Science) 2026. Latest PHE 014 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU BEGC-131 (BAG) 2025-26 Assignment is for January 2026 Session: 30th September, 2026 (for December 2025 Term End Exam). Semester Wise January 2025 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2025 Session: 30th September, 2025 (for December 2025 Term End Exam).
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PHE 14 2025 - English

Tutor Marked Assignment

MATHEMATICAL METHODS IN PHYSICS-III

Course Code: PHE-14

Assignment Code: PHE-14/TMA/2025

Max. Marks: 100

Note: Attempt all questions. Symbols have their usual meanings. The marks for each question are indicated against it.

1. a) Determine the eigenvalues and eigenvectors of the following matrix A:

$A=egin{bmatrix}1&1&1\1&2&1\3&2&3\end{bmatrix}$

b) Show that the eigenvectors belonging to distinct eigenvalues of a hermitian matrix are orthogonal to each other.

c) For the quadratic equation 2x²+ 4xy-y² = 24, write down the matrix of coefficients and diagonalize it. Recast it in new variables and identify the conic section it represents.

d) Define covariant and contravariant vectors. Show that velocity and acceleration are contravariant vectors.

e) Show that the set of all real numbers excluding zero is a group under multiplication.

2. a) Using the method of residues, evaluate the contour integral,

$int _{C}frac{e^{iz}}{(z^{2}+1)}dz$ where C is defined by |z| < 4.

b) Evaluate the integral

$int_{0}^{2pi}frac{d heta}{2+cos heta}=frac{2pi}{sqrt{3}}$

c) Using the Cauchy-Riemann equations, show that w = log z is analytic when z is not zero.

d) Write Laurent series for the function

$f(z)=frac{1}{z(z+2)^{3}}$

about the pole z =-2.

3. a) Determine the Fourier transform of the function f(t) = e^-|t|. Hence deduce that:

$int_{0}^{infty}frac{dx}{1+x^{2}}=frac{pi}{2}$

b) Using the definition determine the Laplace transform of the function f(t) = e ^3t if it exists.

c) Using the method of Laplace transform, solve the following initial value problem:

y"-10y'+9y=5t; y(0)=-1, y'(0) = 2

4. a) Obtain the first three terms in the expansion of the following function in a series of the form $sum_{k=0}^{infty}A_{k}P_{k}(x)$ :

Image ignouassignments-ignouacademy-com-ignou-phe-14-solved-assignment-html-p-assignment-64895

b) Using the following expression for a Bessel function of order m:

$j_{m}(x)=sum_{k=0}^{infty}(-1)^{k}frac{1}{k!Gamma(m+k+1)}left(frac{x}{2} ight)^{2x+m}$

show that

i) $j_{0}(x)=j_{1}(x)$

ii) $frac{d}{dx}left[xJ_{1}(x) ight]=xJ_{0}(x)$

c) Using Rodrigue’s formula, obtain the expression for L₃ (x).

PHE 14 (January 2026 - July 2026) - ENGLISH

Tutor Marked Assignment

MATHEMATICAL METHODS IN PHYSICS-III

Course Code: PHE-14

Assignment Code: PHE-14/TMA/2026

Max. Marks: 100

Note: Attempt all questions. Symbols have their usual meanings. The marks for each question are indicated against it.

1. a) i) Define skew-hermitian matrix, orthogonal matrix and unitary matrix with an example of each.

ii) Determine the eigenvalues and eigenvectors of the following matrix A:

$$A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 3 & 2 & 3 \end{bmatrix}$

b) Show that every eigenvalue of a unitary matrix is of unit modulus.

c) Identify the conic section whose equation is

$$5x^2 - 4xy + 5y^2 = 4$

d) Define covariant and contravariant tensors of rank two. Show that the velocity and acceleration are contravariant vectors.

e) Show that the roots of the equation $z^4 - 1 = 0$ form a cyclic group of order 4.

2. a) Show that the function $\omega = \sin z$ satisfies the Cauchy-Riemann and Laplace equations.

b) Evaluate the integral

$$\oint_C \frac{5z^2 - 3z + 2}{(z - 1)^3} dz$
where C is any simple closed curve enclosing $z = 1$ .

c) Obtain the Laurent series expansion of the function $\frac{e^z}{(z-2)^2}$ about the singularity $z = 2$ .

d) Using the method of residues, show that:

$$\int_{0}^{\infty} \frac{dx}{x^2 + 1} = \frac{\pi}{2}$

3. a) Determine the Fourier transform of the function :

$$f(t) = \begin{cases} \sin 3t & -\pi \le t \le \pi \\ 0 & \text{otherwise} \end{cases}$

b) Determine the Laplace transform of $f(t) = t^2 e^{2t}$ .

c) Using the method of Laplace transforms, solve the following initial value problem:

$$y'' - 3y' + 2y = 6e^{-t}; \quad y(0) = 3, \quad y'(0) = 3$

4. a) Show that P₁(x) is orthogonal to [P_n(x)]² on the interval (-1, 1).

b) Show that

$$J_0(x) - J_2(x) = 2 \frac{d}{dx} [J_1(x)]$
c) Expand the function $f(x) = x^4 - 1$ in a series of the form $\sum_{k=0}^{\infty} A_k P_k(x)$ .

d) Using Rodrigue’s formula, obtain expression for the Hermite polynomial H₃(x) and show that $H_3'(x) = 6H_2(x)$ .

PHE 14 2025 - Hindi

अध्यापक जांच सत्रीय कार्य भौतिकी में गणितीय विधियाँ-III

पाठ्यक्रम कोड: PHE-14

सत्रीय कार्य कोड: PHE-14/TMA/2025

अधिकतम अंक : 100

नोट: सभी प्रश्न हल करें। प्रतीकों के अपने सामान्य अर्थ हैं। प्रत्येक प्रश्न के अंक उसके सामने दर्शाए गए हैं।

1. क) निम्नलिखित आव्यूह 4 के आइगेन मान और आइगेन सदिष ज्ञात कीजिए।

$A=egin{bmatrix}1&1&1\1&2&1\3&2&3end{bmatrix}$

ख) सिद्ध करें कि हर्मिटी आव्यूह के भिन्न आइगेन मानों के संगत आइगेन सदिष एक दूसरे के प्रति लांबिक होते हैं।

ग) द्विघात समीकरण 2x²+4xy-y² = 24, के गुणांकों का आव्यूह लिखिए और उसका विकर्णन कीजिए। उसे नए चरों में लिखिए और बताइए कि यह किस शांकव परिच्छेद को निरूपित करता है।

घ) सहपरिवर्ती सदिष और प्रतिपरिवर्ती सदिष की परिभाषा दें। सिद्ध करें कि वेग और त्वरण प्रतिपरिवर्ती सदिष हैं।

ड.) अवयव शून्य को छोड़कर सभी वास्तविक संख्याओं का समुच्चय गुणन के अधीन एक समूह होता है।

2.क) अवषिष्ट विधि का उपयोग कर, कंटूर समाकल को परिकलित करें :

$oint_{C}frac{e^{iz}}{(z^{2}+1)}dz$ जहां C, |=| < 4 द्वारा परिभाषित है।

ख) निम्नलिखित समाकल को परिकलित कीजिए :

$int_{0}^{2pi}frac{d heta}{2+cos heta}=frac{2pi}{sqrt{3}}$

ग) कौषी-रीमॉन समीकरणों का उपयोग कर सिद्ध करें कि w = logz विष्लेषिक है जब शून्य नहीं है।

घ) z=-2 अनंतक के इर्द-गिर्द फलन $f(z)=frac{1}{z(z+2)^3}$ की लौरों श्रेणी लिखें।

3. क) फलन f(t) = e ^-|t| का फूरिए रूपांतर प्राप्त करें। इससे सिद्ध करें कि :

$int_{0}^{infty}frac{dx}{1+x^{2}}=frac{pi}{2}$

ख) यदि फलन f(t) = e^3t के लाप्लास रूपांतर का अस्तित्व है, तो परिभाषा का प्रयोग करते हुए उसका लाप्लास रूपांतर ज्ञात करें।

ग) लाप्लास रूपांतर लागू करके आदि मान समस्या :

$y''-10y'+9y=5x;quad y(0)=-1,quad y'(0)=2$

का हल करें।

4.क) निम्नलिखित फलन :

$f(x)=egin{cases}cos x& ext{for};0leq xleqpi/2\0& ext{for};pi/2leq xleqpiend{cases}$

का प्रसार $sum_{k=0}^{infty}A_{k}P_{k}(x)$ के रूप की श्रेणी में करते हुए उसके प्रथम तीन पद प्राप्त करें।

ख) कोटि के बेसल फलन के लिए निम्नलिखित व्यंजक का प्रयोग करते हुए

$J_{m}(x)=sum_{k=0}^{infty}(-1)^{k}frac{1}{k!Gamma(m+k+1)}left(frac{x}{2} ight)^{2k+m}$

सिद्ध करें कि :

i) j₀ (x) = j₁(x)

ii) $frac{d}{dx}[xJ_{1}(x)]=xJ_{0}(x)$

ग) रोड्रिगेज़ सूत्र की सहायता से L₃(x) का व्यंजक प्राप्त करें।

PHE 14 (January 2026 - July 2026) - HINDI

अध्यापक जांच सत्रीय कार्य

भौतिकी में गणितीय विधियाँ - III

पाठ्यक्रम कोड : PHE-14

सत्रीय कार्य कोड : PHE-14/TMA/2026

अधिकतम अंक : 100

नोट : सभी प्रश्न हल करें। प्रतीकों के अपने सामान्य अर्थ हैं। प्रत्येक प्रश्न के अंक उसके सामने दिए गए हैं।

1. क) i) विषम-हर्मिटी आव्यूह, लांबिक आव्यूह और ऐकिक आव्यूह को उदाहरण सहित परिभाषित कीजिए।

ii) निम्नलिखित आव्यूह A के आइगेन मान और आइगेन सदिश प्राप्त कीजिए :

$$A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 3 & 2 & 3 \end{bmatrix}$

ख) सिद्ध कीजिए कि ऐकिक आव्यूह का प्रत्येक आइगेनमान एकक मापांक वाला होता है।

ग) उस शंकु परिच्छेद को पहचानिए जिसका समीकरण $5x^2 - 4xy + 5y^2 = 4$ है।

घ) कोटि दो वाले सहपरिवर्ती और प्रतिपरिवर्ती टेन्सरों को परिभाषित कीजिए। सिद्ध करें कि वेग और त्वरण प्रतिपरिवर्ती सदिश हैं।

ङ) सिद्ध कीजिए कि समीकरण $z^4 - 1 = 0$ के मूल, कोटि 4 का चक्रीय समूह बनाते हैं।

2. क) सिद्ध कीजिए कि फलन $\omega = \sin z$ कौशी-रीमान और लाप्लास समीकरणों को संतुष्ट करता है।

ख) निम्नलिखित समाकल को परिकलित कीजिए :

$$\oint_{C} \frac{5z^2 - 3z + 2}{(z - 1)^3} dz$

जहां $C, z = 1$ को परिबद्ध करने वाला एक साधारण बंद वक्र है।

ग) विचित्रता $z = 2$ के प्रति फलन $\frac{e^z}{(z - 2)^2}$ का लौराँ श्रेणी प्रसार प्राप्त कीजिए।

घ) अवशिष्ट विधि का उपयोग कर, सिद्ध करें कि :

$$\int_{0}^{\infty} \frac{dx}{x^2 + 1} = \frac{\pi}{2}$

3. क) निम्नलिखित फलन का फूरिये रूपांतर निर्धारित करें :

$[f(t)=egin{cases}sin 3t,&-pile tlepi,\0,& ext{otherwise}.end{cases}]$

ख) फलन $f(t) = t^2 e^{2t}$ का लाप्लास रूपांतर निर्धारित करें।

ग) लाप्लास रूपांतरण विधि का प्रयोग करके निम्नलिखित आदि मान समस्या का हल प्राप्त करें :

$$y'' - 3y' + 2y = 6e^{-t}; \quad y(0) = 3, \quad y'(0) = 3$

4. क) सिद्ध करें कि P_l(x), अंतराल (-1, 1) में P_k(x) के लांबिक है।

ख) सिद्ध करें कि :

$$J_0(x) - J_2(x) = 2 \frac{d}{dx} [J_1(x)]$
ग) फलन $f(x) = x^4 - 1$ का प्रसार $\sum_{k=0}^{\infty} A_k P_k(x)$ के रूप की श्रेणी में करें।

घ) रोड्रिगेज़ सूत्र का प्रयोग करके हर्मिट बहुपद H₄(x) का व्यंजक प्राप्त करें और सिद्ध करें कि $H_3'(x) = 6 H_2(x)$ ।

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