gamm

Gamm, often referred to as the "gamma function," is a fundamental concept in mathematics and statistics. It is an extension of the factorial function to real and complex numbers. While the factorial function is only defined for non-negative integers (n!), the gamma function, denoted by the Greek letter gamma (Γ), can be evaluated for any complex number except for non-positive integers.

The gamma function is formally defined by an integral: Γ(z) = ∫₀^∞ t^(z-1)e^(-t) dt, where the integral converges for complex numbers z with a positive real part. A key property of the gamma function is its relationship with the factorial: Γ(n) = (n-1)! for any positive integer n. This means Γ(1) = 0! = 1, Γ(2) = 1! = 1, Γ(3) = 2! = 2, and so on.

Another important identity is the recurrence relation Γ(z+1) = zΓ(z). This relation allows for the computation of the gamma function for any complex number, provided its value is known for a number with a smaller real part. The gamma function appears in various fields, including calculus, probability theory, number theory, and physics, often in the context of integrals, series, and probability distributions like the gamma distribution. Its ability to generalize factorials makes it indispensable for advanced mathematical analysis and applications.