1. Functions of a complex variable
(a) Elementary properties of analytic functions
(b) Integration in the complex plane
(c) Analytic functions
(d) Calculus of residues: applications
(e) Periodic functions: Fourier series
(f) Gamma function
2. Differential Equations: analytical methods
(a) Linear differential equations and their power series solutions
(b) Legendre’s equation
(c) Bessel’s equation
(d) Hypergeometric equation
3. Hilbert Spaces
(a) Infinite-dimensional vector spaces
(b) Function spaces
(c) Fourier series
(d) Fourier integral and integral transforms
(e) Orthogonal polynomials
4. Partial differential equations
(a) Linear first-order equations
(b)The Laplacian and the Green function for Laplace’s equation
(c) Time-dependent partial differential equations: The diffusion
equation and the Schrödinger equation
(d) Nonlinear partial differential equations and solitons