The fundamental goal is to cover those mathematical theories essential to the study of quantum mechanics and quantitative study of partial differential equations, especially the partial differential equations of mathematical physics (engineering graduate students). The course encompasses the following topics: uniform convergence, Picard's existence proof, Power series techniques, regular singular point theory, Bessel's equation, Legendre, Hermite and Chebychev polynomials, Orthogonal Functions, completeness, convergence in the mean, Sturm-Liouville theory, eigenvalues, eigenfunction expansions, Sturm comparison and oscillation theorems. Separation of variable techniques for the heat, wave, and Laplace's equation.