The gamma function extends the factorial to non-integers: Γ(5) = 4! = 24, and Γ(½) = √π. It appears in probability distributions (gamma, beta, chi-square, Student’s t), in combinatorics and in physics.
How to use the Gamma Function Calculator
- Argument x
Results update as you type, and everything is calculated in your browser.
How it works
The calculator uses the Lanczos approximation for ln Γ(x) and the reflection formula Γ(x)Γ(1 − x) = π / sin(πx) for negative arguments. Γ has poles at 0, −1, −2, …, where it is undefined.
What each result means
- Γ(x) = gamma([Argument x])
- ln |Γ(x)| = ln(abs(gamma([Argument x])))
- Equivalent factorial (x − 1)! = gamma([Argument x])
- Γ(x + 1) = x·Γ(x) = [Argument x]×gamma([Argument x])
Example values
How the main result changes when Argument x changes and everything else stays at the example values above:
| Argument x | Γ(x) |
|---|---|
| 2.75 | 1.6084 |
| 4.13 | 7.0355 |
| 5.5 | 52.3428 |
| 6.88 | 570.4129 |
| 8.25 | 8,376.5124 |
| 11 | 3,628,800 |
Formula
Γ(x) = ∫₀^∞ t^(x−1) e^(−t) dt
Γ(n) = (n − 1)! for whole n, Γ(x + 1) = x·Γ(x), Γ(½) = √π
Worked example
Inputs — Argument x: 5.5.
Results — Γ(x) = 52.3427777846; ln |Γ(x)| = 3.9578139676; Equivalent factorial (x − 1)! = 52.3427777846; Γ(x + 1) = x·Γ(x) = 287.885277815.
Key terms
- Gamma function
- The continuous extension of the factorial.
- Pole
- A point where a function grows without bound.
Frequently asked questions
What is Γ(1/2)?
√π ≈ 1.7724539.
How is the gamma function related to factorials?
Γ(n) = (n − 1)! for positive integers n.
Why is Γ undefined at 0 and negative integers?
The integral diverges there; Γ has simple poles at 0, −1, −2, …
Can Γ(x) be negative?
Yes, between consecutive negative integers, for example Γ(−0.5) = −2√π.