# Gamma Function Calculator

URL: https://speedycalculator.com/calculators/math/gamma-function/
Last updated: 2026-10-30

> The gamma function calculator evaluates Γ(x) for any real x that is not a non-positive integer, along with ln Γ(x) and the factorial value (x − 1)! for whole numbers.

## Inputs

- Argument x

## Results

- Γ(x)
- ln |Γ(x)|
- Equivalent factorial (x − 1)!
- Γ(x + 1) = x·Γ(x)

## 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
```

## Explanation

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 xΓ(x)2.751.60844.137.03555.552.34286.88570.41298.258,376.5124113,628,800

## 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.

## FAQ

**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.


## Sources

- [NIST DLMF – Chapter 5: Gamma function](https://dlmf.nist.gov/5)
- [Wolfram MathWorld – Gamma function](https://mathworld.wolfram.com/GammaFunction.html)