Chapter 7: Special functions

Several Eulerian integral functions have been defined by Legendre, Riemann, Jacobi and other mathematicians, they are related to the function Gamma and their recursive properties rely on integrations. Some of them are detailed in the first section which originates from Legendre (1825), he defined the functions I and L and calculated tables of the numerical values of the function Beta. Here we present them with simple proofs and different methods to get their numerical values. Expansions of other functions solutions of second order differential equations are explicited, in particular the Airy, Bessel, Hermite and Laguerre functions. Byerly (1893) proved result about Bessel's functions and the results about the other equations are new.