## Course Detail

 Course Name Mathematical Physics II Course Code 22PHY503 Semester 7 Credits 4

### Syllabus

##### Unit 1

Complex Variables I
Learning Objectives
Analysis of functions in complex plane

Complex numbers, Roots, Functions of a complex variable, Differentiation of a complex function, Cauchy-Riemann conditions, Analytic functions, Harmonic functions, Special Analytical functions, Multivalued functions and branch cuts, Singularities, and zeros of complex functions

##### Unit 2

Complex Variables – II
Learning Objectives
Theorems on Complex Integration, Taylor and Laurant Series, Residue Theorem
Evaluation of complex and real integrals using Cauchy‘s theorem and Residue theorem.

Complex integrals, Contour integrals, Darboux inequality, Cauchy’s theorem, Cauchy’s integral formula, Derivatives of analytic functions, Taylor and Laurent series, Uniqueness and Convergence. Poles, Residues at Poles, Residue Theorem, Evaluation of integrals using the Residue Theorem, Jordan’s lemma, Application of Residue Theorem. Applications of Complex variables.

##### Unit 3

Ordinary Differential Equations (ODE) – Series Solution
Learning Objectives
Study of second order ordinary differential equations with special emphasis to Series solution method

Basics of series and first order ODE, Second-order linear ordinary differential equations, Ordinary and singular points, Series solution: Frobenius Method, second solution, the Wronskian method, the derivative method, series form of the second solution, Polynomial solution, Solutions of Legendre, Bessel equations etc. and properties.

##### Unit 4

Partial differential equations
Learning Objectives
Study of partial differential equations in rectilinear and curved coordinates with special importance to PDEs of physically relevant systems.
Familiarity with Orthogonal functions and Sturm-Lioville theory

Partial differential equations (PDEs) in Physics: Laplace, Poisson, Helmholtz equations, treatment in curvilinear coordinates. Other PDEs of Mathematical Physics: diffusion and wave equations, Separation of variables and other methods, Applications.

##### Unit 5

Sturm-Liouville theory
Learning Objectives
Study of Sturn-Lioville problem and its use in theoretical physics, Green‘s function techniques and familiarity with basics of group theory.

Sturm-Liouville Problem and its usage in Physics, Problems with Cylindrical symmetry: Bessel functions, Problems with Spherical Symmetry- Spherical Harmonics, Classical Orthogonal Polynomials.
Introduction to Green’s function: Introduction to Green’s function, Properties, Green’s function in one-dimension, Application in differential equations, Eigen function expansion.
Elements of Group theory: Definition, Cyclic groups, group multiplication table, Isomorphic group, Representation, Special groups: SU(2), O(3).

### Objectives & Outcomes

Course Objectives: The purpose of the course is to introduce students to the methods of mathematical physics and to develop required mathematical skills to solve advanced problems in theoretical physics.

Course Outcomes
After completing the course, the student should be able to
CO1. Understand mathematical methods used in various advanced physics courses and apply the techniques in solving problems involved
CO2. Understand the theory of complex functions, with conditions, theorems related to Complex differentiation and Integration and apply them in solving various types of real and complex integrals
CO3. Analyse and solve second order ordinary differential equations using Series solution method etc.
CO4. Sturm-Liouville Problem and Green’s functions and its usage in Physics, solutions of differential equations in rectilinear and curved coordinates with special importance to PDEs of physically relevant systems; introduction to group theory.

Skills: Problem solving skills using various mathematical methods. Mathematical outlook to physical problems.

CO-PO Mapping

 COs PO1 PO2 PO3 PO4 PO5 PSO1 PSO2 PSO3 CO1 3 3 3 3 CO2 3 3 3 3 CO3 3 3 2 3 3 CO4 3 3 3 3

### Text Books & References

Text Books

1. K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical Methods for Physics and Engineering, Cambridge
University Press, 3rd Edition, 2018.
2. G. Arfken, H. Weber and F.E. Harris, Mathematical Methods for Physicists, Elsevier Indian Reprint, 7th
Edition, 2012.

References

1. M.L.Boas, Mathematical Methods in Physical Sciences, Wiley, 3rd Edition, 2006.
2. J. Mathews and R.L. Walker, Mathematical Methods of Physics, Pearson India, 2nd Edition, 2004.
3. C. W. Wong, Introduction to Mathematical Physics: Methods & Concepts, Oxford, 2nd Edition, 2013.

### Evaluation Pattern

 Assessment Internal External Semester Periodical 1 (P1) 15 Periodical 2 (P2) 15 *Continuous Assessment (CA) 20 End Semester 50

*CA – Can be Quizzes, Assignments, Projects, and Reports.

Justification for CO-PO Mapping

 Mapping Justification Affinity level CO1-PO1 CO1 is related to understanding and applying various mathematical techniques and applying them to solve physical problems. This improves student’s knowledge in Mathematics and Physical sciences as any physical situation can be described as a mathematical problem, and hence the affinity level is 3. 3 CO1-PO2 Since PO2 is related to problem analysis and analytical skills and CO1 is related to understanding and applying various mathematical techniques to study physical systems, to assist it; hence the affinity level is maximum ie, 3. 3 CO2-PO1 CO2 is related to study and application of complex analysis and hence the affinity level 3. 3 CO2-PO2 CO2 deals with many analytical techniques of complex analysis and it enhances students analysis and analytic skills, so the affinity 3. 3 CO3-PO1 CO3, related with solutions of second order differential equations using series solution method with application to physically relevant systems and hence CO3 has maximum affinity 3 when mapped with PO1. 3 CO3-PO2 CO3 is related to solving and applications of second order differential equations, resulting in enhancement of students analysis and analytic skills; hence the affinity 3. 3 CO4-PO1 CO4 introduces many advanced topics of mathematical physics, which are very essential to understand any physical system and thus CO4 has maximum affinity of 3 with PO1. 3 CO4-PO2 CO4 involves application of many advanced mathematical methods to solve physically relevant problems and thus CO4 has maximum affinity of 3 with PO2. 3 CO1-PSO1 PSO1 is related to the proficiency in mathematical physics and other theoretical physics topics; which is the aim of this course, as shown by CO1 and hence the affinity is maximum (3). 3 CO1-PSO2 PSO2 involves imparting analytical skills and the CO matches with maximum affinity 3 CO2-PSO1 CO2 enhances knowledge of vast field of complex analysis and it has arious application in other theoretical courses; hence maximum affinity is seen with PSO1 3 CO2-PSO2 PSO2 involves imparting analytical skills and the CO matches with maximum affinity 3 CO3-PSO1 CO3 is related with analysing and solving second order ordinary differential equations; which is a very essential tool solving physically relevant systems; so maximum affinity is seen with PSO1 3 CO3-PSO2 PSO2 involves imparting analytical skills and the CO matches with maximum affinity 3 CO4-PSO1 CO4 covers various advanced mathematical techniques and its application to physically relevant systems that one encounters in theoretical physics. Hence there is a maximum affinity with PSO1 3 CO4-PSO2 PSO2 involves imparting analytical skills and the CO matches with maximum affinity 3

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