Fourier series and Fourier transforms, Laplace transform, application of transform methods to the solution of elementary differential equations of interest in physics and engineering. Partial differential equations, solution of boundary value problems of partial differential equations by various methods, which include separation of variables, the method of integral transforms. Sturn-Liouville theory, uniqueness of solutions. Calculus of residues and applications to the evaluation of integrals and summation of series. Applications to various physical situations, which may include electromagnetic theory, quantum theory, and diffusion phenomena.