Syllabus

Vectors and Matrices: Linear Vector Space, linear independence, linear operator, matrix operations. Gauss-Jordan matrix inverse, system of linear equations. Transformations of coordinates, vectors, linear operators, linear scalar function, metric. Diagonalization. Dirac Notation. Simultaneous Diagonalization. Jordan Canonical forms. Application (periodic spring system) and Hilbert space. 

Tensor and its transformations. Gradient, Divergence, Curl, Laplacian. Integration of vectors- Gauss theorem, Green’s theorem, Stoke’s theorem, Helmholtz theorem. Vector analysis in curvilinear coordinates. 

Infinite Series: Examples. Cauchy criteria for convergence. Ratio test. Root test. Harmonic sum. By integration. Riemann Zeta function. Hypergeometric function. Legendre polynomial. Leibnitz test. Bernoulli’s numbers. Transformations to make a series more convergent. Poisson resummation. Asymptotic error function. Asymptotic series. Uniform and absolute convergence. 

Infinite products: Convergence. Upper and lower bounds. Product representations. Iterative maps. 

Complex Analysis: Mapping, Riemann sheets, branch points. Sterographic projection. Differentiation of complex function, Cauchy-Riemann condition, Analytic function. Integration of complex function. Cauchy integral theorem. Cauchy integral formula. Morera’s theorem. Cauchy’s inequality. Liouville’s theorem. Fundamental theorem of algebra. 

Laurent expansion. Residue theorem. Cauchy principal value of an integral. Pole expansion of meromorphic function (Mittag-Leffler theorem). Product expansion. Rouche’s theorem. 

Evaluation of Integrals: Integration using Symmetry arguments. Schuringer’s trick, Feynman’s trick. Integration by Contours, Jordan’s lemma. Integrals with Branch cuts. Bernoulli’s numbers and Zeta function. Somerfield Watson transformation. 

Saddle point approximation: Gamma function, Modified Bessel function, Complex domain. Analytic continuation: Gamma function, Zeta function. 

Differential equation: First order, integrating factor, scaling. Higher order differential equation, complementary function, particular integral. General method for PI. Method of Variation of parameter for PI. Green’s function. Wronskian. 

Power series method. Legendre equation. Bessel’s equation. QHO, Hermite differential equation. Associated Legendre equation. Strum Liouville problem. 

Recommended Texts

• Mathews and Walker - Mathematical Methods of Physics. (Most of this course is done from this book)

• Arfken, Weber, Harris - Mathematical Methods for Physicists. (Many topics are covered in this book. It’s a comprehensive book for mathematical physics.)

Other Texts

• Whittaker and Watson, A Course of Modern Analysis, Cambridge University Press, 2021.

Similar Courses

• Linear Algebra by Gilbert Strang - https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/. (a more detailed course for the first part of this course)

• Selected Topics in Mathematical Physics, IIT Madras by Prof. V. Balakrishnan - https://nptel.ac.in/courses/115106086. (the complex analysis part only)

All Files