The first two functions discussed in this chapter are due to Euler. The third is usually associated with Riemann, although it was also studied earlier by Euler. Collectively they are of great importance historically, theoretically, and for purposes of calculation.
Historically and theoretically, investigation of these functions and their properties has provided considerable impetus to the study and understanding of fundamental aspects of mathematical analysis, including limits, infinite products, and analytic continuation. They have also motivated advances in complex function theory, such as the theorems of Weierstrass and 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 in terms 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 a point of doing 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 function and their basic properties, we turn to a number of important identities and representations of the gamma function and its reciprocal. Two characterizations of the gamma function are established, one based on complex analytic properties, the other based on a geometric property. Asymptotic properties of the gamma function are considered in detail. The psi function and the incomplete gamma function are introduced.
The identity that evaluates the beta integral in terms of gamma functions has important modern generalizations due to Selberg and Aomoto. Aomoto's proof is sketched.
The zeta function, its functional equation, and Euler's evaluation of ζ(n) for n = 2, 4, 6, …, are the subject of the final section.
The gamma and beta functions
The gamma function was introduced by Euler in 1729 [119] in answer to the question of finding a function that takes the value n! at each nonnegative integer n. At that time, a “function” was understood as a formula expressed in terms of the standard operations of algebra and calculus, so the problem was not trivial.