Chapter 4 - Special functions

Abstract

In this chapter are studied the properties of most of the important special functions of mathematical physics and chemistry. Some of these functions are obtained as solutions of differential equations of the type mentioned in the previous chapter, like Hermite, Laguerre, Legendre, hypergeometric and confluent hypergeometric functions, and different kinds of Bessel’s functions (of integral or half-integral order, spherical and modified Bessel’s functions). Other special functions are those defined by integrals, like the Dirac’s delta function, the gamma function, the incomplete gamma function, the exponential integral function, the generalized exponential integral function, and many auxiliary functions which arise in the calculation of molecular integrals. The chapter introduces also the study of Fourier and Laplace tranforms, the spherical tensors in complex and real form with a glance at the Wigner–Eckart theorem, the generalized spherical tensors, the study of orthogonal polynomials, and, lastly, the Padé approximants and Green’s functions.