What this calculator does
Gamma function works out evaluates the gamma function Γ(x) for any real x, extending the factorial to non-integers. Enter your own figures above and the answer updates as you type: nothing is fixed in the code, so the result reflects exactly the numbers you supply.
The formula this calculator evaluates is printed under the tool and explained below, so you can check the working by hand or reuse it in a spreadsheet.
The formula
FormulaΓ(x) = ∫₀^∞ tˣ⁻¹e⁻ᵗ dt, with Γ(n) = (n−1)! for positive integers n. Computed via the Lanczos approximation
The inputs explained
| Field | What to enter |
|---|---|
| x | A number. Starts at 5. |
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the answer changes with x
Every other input is held at the calculator’s starting values while x varies. Select any row to load that scenario into the calculator.
How the answer changes with x
| x | Γ(x) | x | Formula used |
|---|---|---|---|
| 2.5 | 1.3293 | 2.500 | Γ(x) = ∫₀^∞ tˣ⁻¹e⁻ᵗ dt, via the Lanczos approximation |
| 3.75 | 4.4230 | 3.750 | Γ(x) = ∫₀^∞ tˣ⁻¹e⁻ᵗ dt, via the Lanczos approximation |
| 5 | 24.0000 | 5.000 | 4! = 24.0000 |
| 7.5 | 1,871.2543 | 7.500 | Γ(x) = ∫₀^∞ tˣ⁻¹e⁻ᵗ dt, via the Lanczos approximation |
| 10 | 362,880.00 | 10.000 | 9! = 362,880.00 |
| 15 | 87,178,291,200 | 15.000 | 14! = 87,178,291,200 |