Department of Physics

10ΕΚ411 Mathematical Physics (postgraduate)

Course webpage:   https://eclass.uoa.gr/courses/PHYS230

Course outline

Course content

• Special functions and orthogonal polynomials. Expansion in eigenfunctions. Laplace, diffusion, Helmholtz, Poisson equations.
• Hilbert transform. Dispersion relations. Advanced complex integration techniques and their applications.
• Saddle point and stationary phase approximations. Asymptotic expansions.
• Green's functions: Construction of Green's functions for the Helmholtz, Poisson, Laplace equations and for the wave equation.
• Problems in Cartesian, spherical and cylindrical coordinates with homogeneous and inhomogeneous boundary conditions. Expansion in orthogonal polynomials. The importance of propagator and Green's function in quantum mechanics.
• Integral equations: Fredholm and Volterra equations. Relation to Green's equations. Exact solution techniques through expansion in orthogonal polynomials and through integral transformations (Fourier, Laplace, Mellin, Hankel). Solving through iterative approaches.