Gamma function

The gamma function is defined by the integral Γ(z) = ∫₀^∞ t^(z−1) e^(−t) dt for complex z with Re(z) > 0. This integral converges absolutely in that half-plane and defines an analytic function. The gamma function satisfies the functional equation Γ(z+1) = z Γ(z), which leads to Γ(n) = (n−1)! for positive integers n. It can be analytically continued to the whole complex plane except for simple poles at non-positive integers. The reflection formula Γ(z) Γ(1−z) = π / sin(πz) relates values at z and 1−z, and the duplication formula Γ(z) Γ(z+1/2) = 2^(1−2z) √π Γ(2z) is another important identity.¹

The gamma function was introduced by Leonhard Euler in 1729 as an infinite product, and later as an integral, to interpolate the factorial function. Adrien-Marie Legendre later named it the gamma function and introduced the notation Γ(z). Carl Friedrich Gauss and Karl Weierstrass contributed to its theory, with Weierstrass giving an infinite product definition. The function has been studied extensively since, with connections to number theory, such as in the functional equation of the Riemann zeta function.

The gamma function is ubiquitous in probability and statistics, appearing in the gamma distribution, chi-squared distribution, and Student's t-distribution. In physics, it appears in quantum mechanics, such as in the normalization of wavefunctions, and in the evaluation of integrals in quantum field theory. In engineering, it is used in signal processing and control theory. It also appears in combinatorics, where it generalizes binomial coefficients to non-integer arguments, and in the theory of special functions, where it is a building block for the beta function and hypergeometric functions.

The gamma function has several less-known properties. The Bohr–Mollerup theorem characterizes it as the unique function that is log-convex and satisfies the functional equation and the normalization Γ(1)=1. The gamma function has no zeros, but it has poles at non-positive integers. The digamma function, the logarithmic derivative of Γ, is used in number theory and in the study of harmonic numbers. The gamma function also appears in the evaluation of the volume of an n-dimensional sphere, and in the theory of modular forms. A notable edge case is that Γ(1/2) = √π, which follows from the reflection formula. The function is also used in the definition of the gamma distribution, which models waiting times in Poisson processes.²

Analytic continuation

A technique to extend the domain of a complex function beyond its original definition.

Factorial

The product of all positive integers up to a given number, denoted n!.

Log-convex

A function whose logarithm is convex.

Poles

Points where a complex function goes to infinity in a specific way.