This course will cover some advanced topics in mathematics, in particular those topics used popularly in engineering and physical sciences. The students will master how to use methods like separation of variables, Fourier series, Fourier transform, and Laplace transform, to solve most problems encountered in engineering and physics quantitatively such as vibrations of string, wave equations (Helmholtz equations and Schrodinger equation), heat diffusion/conduction equation. The students also need to understand how some special functions like Bessel function, Legendre polynomial and Hermit function are related to engineering problems. The content includes: Fourier series and Fourier transform; Laplace transform, partial differential equations: wave equation, heat diffusion equation, Poisson equation; Functions of complex variables; and series solution of ordinary differential equations: Legendre, Bessel, Hermite and Laguerre functions.