The first two of the functions discussed in this chapter are due to Euler. The third is usually associated with Riemann, though it was also studied by Euler. Collectively they are of great importance historically, theoretically, and for purposes of calculation.

Historically and theoretically, study of these functions and their properties provided a considerable impetus to the study and understanding of fundamental aspects of mathematical analysis, including limits, infinite products, and analytic continuation. They also motivated advances in complex function theory, such as the theorems of Weierstrass and of Mittag–Leffler on representations of entire and meromorphic functions. The zeta function and its generalizations are intimately connected with questions of number theory.

From the point of view of calculation, many of the explicit constants of mathematical analysis, especially those that come from definite integrals, can be evaluated by means of the gamma and beta functions.

There is much to be said for proceeding historically in discussing these and other special functions, but we shall not make it a point to do so. In mathematics it is often, even usually, the case that later developments cast a new light on earlier ones. One result is that later expositions can often be made both more efficient and, one hopes, more transparent than the original derivations.

After introducing the gamma and beta functions and their basic properties, we turn to a number of important identities and representations of the gamma function and its reciprocal.