Gamma Function Calculator - Compute Gamma(z) Online

The Gamma function, denoted Gamma(z), is one of the most important special functions in mathematics. It extends the concept of factorial to all complex numbers except the non-positive integers. For any positive integer n, Gamma(n) = (n-1)!, making it a natural generalization of the factorial operation. The function was first introduced by Leonhard Euler in the 18th century and has since become indispensable in areas ranging from pure mathematics to theoretical physics and engineering. For positive real numbers, the Gamma function is defined by the integral Gamma(z) = integral from 0 to infinity of t^(z-1) * e^(-t) dt. This integral converges absolutely for all complex numbers with a positive real part. For other values, the function is defined by analytic continuation. Notably, Gamma(z) has simple poles at z = 0, -1, -2, ... and is analytic everywhere else in the complex plane. The Gamma function satisfies several fundamental identities. The recurrence relation Gamma(z+1) = z*Gamma(z) is perhaps the most important, as it mirrors the factorial recurrence n! = n*(n-1)!. Another key identity is the reflection formula: Gamma(z)*Gamma(1-z) = pi/sin(pi*z), which links values on either side of the real axis. The duplication formula Gamma(z)*Gamma(z+1/2) = sqrt(pi)*2^(1-2z)*Gamma(2z) is also widely used. In practice, the Gamma function appears in probability distributions such as the Gamma distribution and the Beta distribution. It is essential in statistics for expressing the normalisation constants of many continuous distributions. In combinatorics, it generalises binomial coefficients to non-integer arguments. In physics, it arises in quantum mechanics, statistical mechanics, string theory, and the computation of Feynman diagrams. This calculator uses the Lanczos approximation, which provides extremely high accuracy (typically 15 or more significant digits) for real arguments. The approximation works by expressing Gamma(z+1) as a product involving a rational function with carefully chosen coefficients. It is computationally efficient and is the method of choice in most software libraries, including Python math.gamma and many scientific computing packages. Whether you are a student learning about special functions, an engineer computing integrals, or a statistician working with continuous distributions, this tool provides instant and reliable results.