Methods of Mathematical Physics
This course introduces complex analysis, integral transforms, differential equations, boundary-value problems, and special functions, providing essential mathematical methods for analyzing waves, heat transfer, potential fields, and a broad range of problems in mathematical physics.
Instructor: Hai-Peng Sun
Course description
Methods of Mathematical Physics provides a systematic introduction to analytical methods and mathematical tools widely used in mathematical physics. The course covers complex functions, complex integration, power-series expansions, the residue theorem, Fourier transforms, Laplace transforms, well-posed problems in mathematical physics, separation of variables, series solutions of second-order ordinary differential equations, and spherical functions.
The course begins with the theory of complex functions and develops the use of analytic functions, contour integration, series expansions, and residue calculations in mathematical and physical applications. It then introduces Fourier and Laplace transforms, with emphasis on spectral representations, transformation properties, and their role in solving differential equations.
The later part of the course focuses on mathematical-physics equations and their well-posed problems. It develops separation of variables, boundary conditions, eigenvalue problems, and eigenfunction expansions, followed by power-series solutions of second-order ordinary differential equations, regular singular points, and spherical-function theory. Emphasis is placed on derivation, method selection, verification of conditions, and interpretation of results. The course aims to equip students with the mathematical tools required to analyze and solve problems related to waves, heat conduction, potential fields, and other physical systems.
Prerequisites
Calculus and introductory college physics.
Topics
The course introduces vector calculus, ordinary and partial differential equations, complex analysis, Fourier methods, and special functions.