PH-601   Methods of Mathematical Physics

1. Fourier series: introduction and general properties, convergence of trigonometric series, Gibbs phenomenon, Parseval’s theorem, applications to various phenomena.
2. Integral transform, development of the Fourier integral, Fourier transform, inversion theorems, Fourier transform of derivatives, convolution theorem, momentum representation, transfer functions.
3. Complex arguments in Fourier transforms. Laplace transform, Laplace transform of derivatives, convolution products and Faltung’s theorem, inverse Laplace transform.
4. Partial differential equations. Separation of variables in three dimensions, method of characteristics. Boundary value problems.
5. Integral transforms, generating functions, Neumann series, separable (degenerate) kernels, Hilbert–Schmidt theory, integral equations.
6. Calculus of variations: dependent and independent variables, Euler-Lagrange equation and applications, several independent and dependent variables, Lagrange multipliers, variational principle with constraints, Rayleigh–Ritz variational technique, application to discrete mesh.
7. Nonlinear methods and chaos, the logistic map, sensitivity to initial conditions and parameters, nonlinear differential equations.
8. Probability: definitions and simple properties, random variables, binomial distribution, Poisson distribution, Gauss's normal distributions, statistics.

Recommended texts:

• Mathematical Methods for Physicists, Arfken & Weber (Academic Press, 6th edition, 2005).
• Mathematical Methods for Physicists, Tai L. Chow (Cambridge University Press, 2002).