Fonctions sp\'eciales et polyn\^omes orthogonaux: cours et exercices corrig\'es

TL;DR

This paper provides a comprehensive study of special functions and orthogonal polynomials, essential tools in physics and mathematics, including their properties, applications, and computational aspects, with exercises and solutions.

Contribution

It offers an in-depth French-language course and exercises on special functions and orthogonal polynomials, integrating theoretical foundations with computational tools.

Findings

Detailed coverage of gamma, beta, Bessel, and hypergeometric functions

Exploration of orthogonal polynomials and their properties

Practical exercises with solutions for students

Abstract

This report (written in French) is devoted to studying special functions the most used in physics. Special functions are a very broad branch of mathematics, theoretical physics, and mathematical physics. They appeared in the nineteenth century as solutions of equations in mathematical physics, particularly partial differential equations of order two and four. Their knowledge is essential for the proper handling and understanding of current problems in physics. They are also related to the art of scientific computing in physics and mathematics. Special functions are included in many computer algebra software such as Matlab, Mathematica, and Maple, and students are strongly encouraged to take part in this development which has become indispensable for the treatment of almost all current problems in physics. The manuscript contains six chapters: gamma and beta functions, Bessel functions, Fresnel error, and integral function, exponential integral, sine integral, cosine integral, and logarithm integral, orthogonal polynomials, and finally hypergeometric functions.