**PHY ****601: Advanced Mathematical Methods for Physics**** (4)**

*Learning Objectives*:

The main objective of the course is to equip the students with the tools of mathematics which are required for graduate courses and physics research.

*Course Contents*:

Vector space and matrices, linear independence, bases dimensionality, Inner product, tensors, parallel transport, linear transformation matrices, inverse, orthogonal and unitary matrices, independent element of a matrix, Eigen values and Eigen vectors, diagonalization.

Theory of complex variables, Cauchy- Riemann condition, analytic functions, Cauchy’s theorem, Cauchy integral formula, Laurent series, singularities, branch points and cuts, residue theorem, contour integration, evaluation of definite integrals, method of steepest descent.

Ordinary differential equations, second order linear ODEs with variable coefficients, Solution by series expansion, non-homogeneous differential equations and solution by the method of Green’s functions with applications. Eigenvalue methods, up to Strum-Liouville systems. Special functions, Legendre, Bessel, Hermite and Laguerre functions with their physical applications, generating functions, orthogonality conditions, recursion relations, Legendre, Bessel, Hermite, Laguire equations and their solutions. Fourier integral and transforms, inversion theorem, Fourier transform of derivatives, convolution theorem.

Partial differential equations, Solution of Laplace and Poisson's equations, Wave equation

Introduction to Groups, Representations, Finite Groups, Permutation Groups, Continuous Groups, Lie algebras.

*Suggested Books*:

- B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 6th Ed.
- P. K. Chattopadhyay, Mathematical Physics.
- M. L. Boas, Mathematical Methods in Physical Sciences.
- S. D. Joglekar, Mathematical Physics: The Basics.
- A. K. Ghatak, Mathematical Method of Physics.
- F. B. Hildebrand, Methods of Applied Mathematics.
- A. W. Joshi, Elements of Group Theory for Physicist.
- S. Hassani, Mathematical Physics.
- P. Dennery and A. Krzywicki, Mathematics for Physicists.
- J. Mathews and R. L. Walker, Mathematical Methods of Physics.

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