Gamma Function Calculator (Factorial of Any Number, Beta Function)
Γ(x), the factorial of non-whole numbers, ln Γ(x) and the beta function B(x, y), to about 15 significant figures.
Results
The dashed line marks the value you entered.
What this tool does
The gamma function stretches the factorial to every positive number, so that 4.5! has a meaning: Γ(x + 1) = x!, and for whole numbers Γ(n) = (n − 1)!. It satisfies Γ(x + 1) = x Γ(x), and Γ(½) = √π ≈ 1.7725 is the best-known fractional value. It turns up in the t, chi-square and gamma distributions, and the beta function B(x, y) = Γ(x)Γ(y) ÷ Γ(x + y) is the normalising constant of the beta distribution. The values are computed with the Lanczos approximation; Γ grows so fast that results above x ≈ 170 overflow, which is why the input stops there.
Formula
Γ(x) = ∫₀^∞ tˣ⁻¹ e⁻ᵗ dt · Γ(n) = (n − 1)! · Γ(x + 1) = x Γ(x) · Γ(½) = √π · B(x, y) = Γ(x) Γ(y) ÷ Γ(x + y)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
x | x (greater than 0) | — |
y | y (for the beta function) | — |
G | Γ(x) | — |
F | x! = Γ(x + 1) | — |
L | ln Γ(x) | — |
B | B(x, y) = Γ(x) Γ(y) ÷ Γ(x + y) | — |
Worked example
- x (greater than 0)4.5
- y (for the beta function)2.5
- Γ(x)11.63172840
- x! = Γ(x + 1)52.34277778
- ln Γ(x)2.45373657
- B(x, y) = Γ(x) Γ(y) ÷ Γ(x + y)0.02147573
Limitations
- The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.
- The default values are typical reference figures, not measurements of your situation. Replace them with your own data whenever you have it.