Math 316

Mathematical Methods (MATH316, 4 credits)

[1] Inner product space. Vector space; inner product space; the space of square integrable functions L²; convergence in L²; orthogonal functions (9 lectures).
[2]. The Sturm-Liouville theory. Linear second-order equations; self-adjoint differential operator; the Sturm

Liouville problem; existence and completeness of the eigenfunctions; regular and singular problems (9 lectures).

[3]. Fourier series. The fundamental theorem of Fourier series in L²; pointwise theory of Fourier series; applications to
boundary-value problems (6 lectures).

[4]. Orthogonal polynomials. Legendre, Hermite, and Laguerre polynomials as solutions of certain singular Sturm-Liouville problems; their orthogonality and completeness properties; generalized Fourier expansions (9 lectures).

[5]. Bessel functions. The gamma function; Bessel's equation and Bessel functions of the first kind; orthogonality properties (6 lectures).

[6]. Fourier transformation. The fourier transform; the Fourier integral; properties and applications in PDE (6 lectures).