Gamma Function Calculator: Γ(x) for Any Real Number, with Exact Half-Integer Values

How it works

Γ(x) = ∫₀^∞ t^(x−1) e^(−t) dt (x > 0) · Γ(x + 1) = x·Γ(x) · Γ(n) = (n − 1)! · Γ(½) = √π · Γ(n + ½) = (2n)!/(4ⁿ n!)·√π · Γ(x)·Γ(1 − x) = π / sin(πx) · Lanczos: Γ(x) ≈ √(2π)·(x + g − ½)^(x − ½)·e^−(x + g − ½)·A_g(x)

The gamma function extends the factorial to every real number except 0 and the negative integers: Γ(n) = (n − 1)! for whole numbers, so Γ(5) = 4! = 24, and in between it is a smooth curve. Its key property is the recurrence Γ(x + 1) = x·Γ(x), which lets you step from any argument to its neighbours; Γ(½) = √π then gives exact values for every half-integer, such as Γ(4.5) = (105/16)√π. For negative non-integers Γ alternates in sign between the poles, and the reflection formula Γ(x)·Γ(1 − x) = π/sin(πx) links it to positive arguments. This calculator evaluates Γ with the Lanczos approximation (g = 7, nine coefficients), which is accurate to about 15 significant digits, and uses reflection below ½. Above x ≈ 171.6, Γ(x) no longer fits in a standard floating-point number, so it works with ln Γ(x) and shows the result as m × 10ⁿ (for example Γ(1000) = 4.02 × 10²⁵⁶⁴). You can type fractions like 1/3. Next to Γ(x) you also get ln|Γ(x)|, the reciprocal 1/Γ(x), x! = Γ(x + 1), the digamma function ψ(x), exact factorial or √π forms where they exist, and a table of neighbouring values. The factorial calculator handles whole numbers only, and the gamma distribution calculator gives probabilities of a statistical distribution; this page is for the function Γ(x) itself, as it appears in probability densities, physics and the beta function B(a, b) = Γ(a)Γ(b)/Γ(a + b).

Worked example

For the default x = 4.5, apply Γ(x + 1) = x·Γ(x) four times to reach 0.5: Γ(4.5) = 3.5 × 2.5 × 1.5 × 0.5 × Γ(0.5). The product of the factors is 6.5625 and Γ(0.5) = √π = 1.772453851, so Γ(4.5) = 6.5625 × 1.772453851 = 11.6317284, exactly (105/16)√π. Also ln Γ(4.5) = 2.453736571, 1/Γ(4.5) = 0.08597174606, 4.5! = Γ(5.5) = 52.34277778 and ψ(4.5) = 1.388870926.

Sources: NIST DLMF §5.4: Special values of the gamma function · Omni Calculator: Gamma function (Lanczos and Stirling, half-integer values) · Math is Fun: Gamma function (definition and recurrence) · Redcrab Software: Gamma function calculator (Γ(0.5), Γ(1.5), Γ(2.5))

Frequently asked questions