Course

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).

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

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.

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

Justification for CO-PO Mapping