Chapter 2
Special Functions

The functions of this chapter are called special not because they are unusual but because they are singled out: each arises repeatedly, each fails to be elementary, and each is therefore given a name and tabulated once rather than re-derived. The gamma function extends the factorial to non-integer argument; the beta function does the same for the binomial coefficient and reduces entirely to the gamma; the error function supplies the antiderivative of the Gaussian that elementary calculus cannot; and the incomplete forms of the gamma and beta functions turn out to be the distribution functions of most of classical statistics.

They are gathered here because they form one subject. Each is defined by a definite integral depending on a parameter, each is evaluated by the same techniques of Chapter 1, and the relations between them are as important as the functions themselves.