Special Functions

Gamma, Beta and error functions. Bessel functions of the first and second kind. Linear independence and recurrence relations of them. Modified Bessel functions of the first and second kind. Linear independence and recurrence relations of them. Solving ordinary differential equations in terms of Bessel functions. Lommel's integrals. Roots of Bessel functions. Fourier-Bessel series. Orthogonal polynomials. Three-term recurrence relation. Darboux-Christoffel formula. Roots of orthogonal polynomials. Rodrigues' formula. Generating function. Applications to classical orthogonal polynomials.

Objectives

The ability to handle special functions during anyone's research and use the analysis of these functions to achieve better results in programs such as Maple, Mathematica e.t.c. in PC.

Prerequisites

Ordinary Differential Equations I, Ordinary Differential Equations II

Syllabus

Gamma, Beta and error functions. Bessel functions of the first and second kind. Linear independence and recurrence relations of them. Modified Bessel functions of the first and second kind. Linear independence and recurrence relations of them. Solving ordinary differential equations in terms of Bessel functions. Lommel's integrals. Roots of Bessel functions. Fourier-Bessel series. Orthogonal polynomials. Three-term recurrence relation. Darboux-Christoffel formula. Roots of orthogonal polynomials. Rodrigues' formula. Generating function. Applications to classical orthogonal polynomials.