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Gamma Function Calculator - Online Advanced Factorial

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Gamma Function Calculator

Online Advanced Factorial — Compute Γ(x) for any real number with high precision

Compute Γ(x)
Enter a value and press Calculate
Common Gamma Values
x Γ(x) Note
−2.5−8√π/15 ≈ −0.9453half-int
−1.54√π/3 ≈ 2.3633half-int
−0.5−2√π ≈ −3.5449half-int
0Undefined (pole)—
0.25≈ 3.6256—
0.5√π ≈ 1.7725half-int
11integer
1.5√π/2 ≈ 0.8862half-int
21integer
2.53√π/4 ≈ 1.3293half-int
32integer
3.515√π/8 ≈ 3.3234half-int
46integer
524integer
6120integer
10362880integer

For half-integers, Γ(n+½) = (2n)!√π / (4ⁿ n!)

Frequently Asked Questions
The Gamma function Γ(z) is a meromorphic function that extends the factorial to complex numbers. For positive integers n, Γ(n) = (n−1)!. It was introduced by Leonhard Euler in 1729 and is fundamental in mathematics, physics, and statistics. The integral definition is Γ(z) = ∫₀^∞ tz−1 e−t dt for Re(z) > 0.
For any positive integer n, Γ(n) = (n−1)!. For example, Γ(5) = 4! = 24. This relationship is why the Gamma function is often called the "generalized factorial." The functional equation Γ(x+1) = x·Γ(x) mirrors the factorial recursion n! = n·(n−1)!.
Γ(1/2) = √π ≈ 1.7724538509. This elegant result follows from the Gaussian integral. Using the substitution t = u² in the integral definition yields Γ(1/2) = ∫₀^∞ t−1/2 e−t dt = 2∫₀^∞ e−u² du = √π. This is one of the most celebrated values in analysis.
The Gamma function has simple poles at all non-positive integers: x = 0, −1, −2, −3, … At these points, Γ(x) is undefined (tends to ±∞). Near a pole at x = −n, the residue is (−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 Γ(x) = π / (sin(πx)·Γ(1−x)). For large x, we also compute lnΓ(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), Γ(x) exceeds JavaScript's maximum numeric value (~1.79×10³⁰⁸) and overflows to Infinity. In these cases, lnΓ(x) — the natural logarithm of the Gamma function — remains manageable. For example, lnΓ(200) ≈ 857.9. Our calculator automatically provides lnΓ(x) when the direct value overflows or when it's more informative.
Yes! For negative non-integer x, Γ(x) is well-defined (though it alternates in sign between poles). For example, Γ(−0.5) = −2√π ≈ −3.5449 and Γ(−1.5) = 4√π/3 ≈ 2.3633. Our calculator handles these cases using the reflection formula. The sign pattern for negative half-integers follows (−1)n+1 for Γ(−n+½).
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