11 - The Gamma, Beta, Pi, and Psi Functions

Abstract

This chapter focuses on the gamma, beta, pi, and psi functions. The chapter discusses the Euler integral and limit and infinite product representations for Γ(x). The gamma function, denoted by Γ(x), provides a generalization of factorial n to the case in which n is not an integer. It is defined by a Euler integral. The chapter reviews several special properties of Γ(x), the asymptotic representations of Γ(x) and n!, the gamma function in the complex plane, the psi (digamma) function, and the beta function. The graph of Γ(x) and the tabular values of Γ(x) and ln Γ(x) are also reviewed. The numerical values of Γ(x) and ln Γ(x) for 0 ≤x ≤ 2 are presented in a tabulated form.