Gamma Function Calculator
Our free algebra calculator solves gamma function problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
Gamma(x) = integral from 0 to infinity of t^(x-1) * e^(-t) dt
The gamma function is defined by this improper integral for x > 0 and extended to all complex numbers (except non-positive integers) via analytic continuation. For positive integers, Gamma(n) = (n-1)!. The function satisfies the recurrence Gamma(x+1) = x * Gamma(x).
Worked Examples
Example 1: Gamma of a Positive Integer
Problem:Calculate Gamma(6).
Solution:For positive integers, Gamma(n) = (n-1)!\nGamma(6) = (6-1)! = 5!\n5! = 5 * 4 * 3 * 2 * 1 = 120\nSo Gamma(6) = 120
Result:Gamma(6) = 120
Example 2: Gamma of a Half-Integer
Problem:Calculate Gamma(3.5).
Solution:Using the recurrence Gamma(x+1) = x * Gamma(x):\nGamma(3.5) = 2.5 * Gamma(2.5)\nGamma(2.5) = 1.5 * Gamma(1.5)\nGamma(1.5) = 0.5 * Gamma(0.5)\nGamma(0.5) = sqrt(pi) = 1.7724538509\nGamma(1.5) = 0.5 * 1.7724538509 = 0.8862269255\nGamma(2.5) = 1.5 * 0.8862269255 = 1.3293403882\nGamma(3.5) = 2.5 * 1.3293403882 = 3.3233509705
Result:Gamma(3.5) = 3.3233509705
Frequently Asked Questions
What is the gamma function and what does it do?
The gamma function, denoted as a capital Greek letter gamma, is a generalization of the factorial function to all complex and real numbers (except non-positive integers). For positive integers n, the gamma function satisfies the relation Gamma(n) = (n-1)!, so Gamma(5) = 4! = 24. The key innovation is that while the factorial is only defined for non-negative integers, the gamma function extends this concept smoothly to all real and complex numbers. It was introduced by Leonhard Euler in the 18th century and has become one of the most important special functions in mathematics, appearing throughout probability theory, statistical distributions, complex analysis, combinatorics, and physics.
How is the gamma function related to factorials?
The gamma function is the unique smooth extension of the factorial function to non-integer values. The fundamental relationship is Gamma(n) = (n-1)! for any positive integer n. This means Gamma(1) = 0! = 1, Gamma(2) = 1! = 1, Gamma(3) = 2! = 2, Gamma(4) = 3! = 6, Gamma(5) = 4! = 24, and so on. Notice the offset by one, which is a historical convention. The gamma function also satisfies the recursive property Gamma(x+1) = x * Gamma(x), which mirrors the factorial recurrence n! = n * (n-1)!. This recurrence relation, combined with the initial condition Gamma(1) = 1, uniquely determines the gamma function among log-convex functions by the Bohr-Mollerup theorem.
What is the value of Gamma(1/2) and why is it important?
Gamma(1/2) equals the square root of pi, approximately 1.7724538509. This remarkable result connects the gamma function to the most fundamental constant in mathematics. The proof comes from the integral definition: Gamma(1/2) = integral from 0 to infinity of t^(-1/2) * e^(-t) dt, which through the substitution t = u^2 transforms into 2 times the integral of e^(-u^2) from 0 to infinity, which equals the square root of pi (the Gaussian integral). This value is critically important in probability and statistics because it appears in the normalization constant of the normal distribution. It also enables computing gamma function values at all half-integers using the recurrence relation.
Where does the gamma function appear in probability and statistics?
The gamma function is ubiquitous in probability and statistics, forming the backbone of several major probability distributions. The gamma distribution itself uses the gamma function in its probability density function and is used to model waiting times, rainfall amounts, and insurance claims. The chi-squared distribution, fundamental to hypothesis testing, is a special case of the gamma distribution. The beta function B(a,b) = Gamma(a)*Gamma(b)/Gamma(a+b) defines the beta distribution used in Bayesian statistics. The Student t-distribution, F-distribution, and Dirichlet distribution all involve gamma functions in their formulas. Even the normalizing constant of the multivariate normal distribution uses the gamma function through the relationship between the gamma function and the volume of n-dimensional spheres.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy