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Gamma Function Calculator - Online Advanced Factorial

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Gamma Function Calculator

Online Advanced Factorial β€” Compute Ξ“(x) for any real number with high precision

Compute Ξ“(x)
Enter a value and press Calculate
Common Gamma Values
x Ξ“(x) Note
βˆ’2.5βˆ’8βˆšΟ€/15 β‰ˆ βˆ’0.9453half-int
βˆ’1.54βˆšΟ€/3 β‰ˆ 2.3633half-int
βˆ’0.5βˆ’2βˆšΟ€ β‰ˆ βˆ’3.5449half-int
0Undefined (pole)β€”
0.25β‰ˆ 3.6256β€”
0.5βˆšΟ€ β‰ˆ 1.7725half-int
11integer
1.5βˆšΟ€/2 β‰ˆ 0.8862half-int
21integer
2.53βˆšΟ€/4 β‰ˆ 1.3293half-int
32integer
3.515βˆšΟ€/8 β‰ˆ 3.3234half-int
46integer
524integer
6120integer
10362880integer

For half-integers, Ξ“(n+Β½) = (2n)!βˆšΟ€ / (4ⁿ n!)

Frequently Asked Questions
The Gamma function Ξ“(z) is a meromorphic function that extends the factorial to complex numbers. For positive integers n, Ξ“(n) = (nβˆ’1)!. It was introduced by Leonhard Euler in 1729 and is fundamental in mathematics, physics, and statistics. The integral definition is Ξ“(z) = βˆ«β‚€^∞ tzβˆ’1 eβˆ’t dt for Re(z) > 0.
For any positive integer n, Ξ“(n) = (nβˆ’1)!. For example, Ξ“(5) = 4! = 24. This relationship is why the Gamma function is often called the "generalized factorial." The functional equation Ξ“(x+1) = xΒ·Ξ“(x) mirrors the factorial recursion n! = nΒ·(nβˆ’1)!.
Ξ“(1/2) = βˆšΟ€ β‰ˆ 1.7724538509. This elegant result follows from the Gaussian integral. Using the substitution t = uΒ² in the integral definition yields Ξ“(1/2) = βˆ«β‚€^∞ tβˆ’1/2 eβˆ’t dt = 2βˆ«β‚€^∞ eβˆ’uΒ² du = βˆšΟ€. This is one of the most celebrated values in analysis.
The Gamma function has simple poles at all non-positive integers: x = 0, βˆ’1, βˆ’2, βˆ’3, … At these points, Ξ“(x) is undefined (tends to ±∞). Near a pole at x = βˆ’n, the residue is (βˆ’1)ⁿ/n!. Our calculator detects these poles and warns you accordingly.
We implement the Lanczos approximation with g=7 and 9 coefficients, which achieves ~12–14 decimal digits of precision for real arguments. For x < 0.5 (and not at a pole), we apply the reflection formula Ξ“(x) = Ο€ / (sin(Ο€x)Β·Ξ“(1βˆ’x)). For large x, we also compute lnΞ“(x) to avoid overflow.
The Gamma function appears throughout science and engineering: probability theory (Gamma and Chi-squared distributions), combinatorics (binomial coefficients with non-integer arguments), physics (quantum mechanics, statistical thermodynamics), fractional calculus, and the Riemann zeta function. It's also essential in computing volumes of n-dimensional spheres.
For large x (roughly x > 170), Ξ“(x) exceeds JavaScript's maximum numeric value (~1.79Γ—10³⁰⁸) and overflows to Infinity. In these cases, lnΞ“(x) β€” the natural logarithm of the Gamma function β€” remains manageable. For example, lnΞ“(200) β‰ˆ 857.9. Our calculator automatically provides lnΞ“(x) when the direct value overflows or when it's more informative.
Yes! For negative non-integer x, Ξ“(x) is well-defined (though it alternates in sign between poles). For example, Ξ“(βˆ’0.5) = βˆ’2βˆšΟ€ β‰ˆ βˆ’3.5449 and Ξ“(βˆ’1.5) = 4βˆšΟ€/3 β‰ˆ 2.3633. Our calculator handles these cases using the reflection formula. The sign pattern for negative half-integers follows (βˆ’1)n+1 for Ξ“(βˆ’n+Β½).
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