ABSTRACT
In mathematics you don’t understand things. You just get used to them.
– John von Neumann
Special functions do not have any precise definition. But they constitute a class of non-elementary functions,* many of which arise in the course of solution of certain ordinary differential equations (ODEs) with variable coefficients. Special functions are expressed in the form of infinite series or integral functions. In fact, many of the special functions were developed in the course of solution of differential equations arising out of various kinds of physical problems. There is no comprehensive list of special functions, but their number may well be above 50. Some typical definite integrals can also be reduced to special functions. The common special functions that are of interest in chemical engineering include the gamma, beta and error functions (these three special functions and several others are expressed in the form of integrals), Bessel functions, Legendre functions and hypergeometric functions. There are certain special functions that arise in quantum mechanics, for example the Airy functions and Hermite polynomials. Selected special functions, their properties, how these functions appear as solutions of certain second-order ODEs with variable coefficients, as well as their applications, are discussed in this chapter.
