316 Math (Sturm-Liouville theory)

1Inner product space.
Vector space; inner product space; the space of square integrable functions L²convergence 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).

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