Gamma Function
The Gamma function extends the factorial to real and complex numbers.
Learn the formula, key properties, and worked examples.
The Formula
Key property: Γ(n) = (n−1)! for positive integers
The Gamma function, introduced by Swiss mathematician Leonhard Euler in the 1720s, extends the concept of the factorial to all real and complex numbers — not just positive integers. While n! is only defined for non-negative integers, Γ(n) fills in the gaps, giving meaningful values for numbers like Γ(1.5) or Γ(0.5).
The key relationship is Γ(n) = (n−1)! for positive integers. So Γ(5) = 4! = 24, Γ(6) = 5! = 120, and so on. The function shifts the argument by 1 compared to the ordinary factorial, which is a historical convention from Euler's original definition.
One of the most remarkable values is Γ(1/2) = √π ≈ 1.7725. This identity shows up throughout physics and statistics, particularly in the normalization of the Gaussian (normal) distribution. The functional equation Γ(n+1) = n × Γ(n) allows values to be computed recursively from known ones.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| Γ(n) | Gamma function evaluated at n | dimensionless |
| n | Input value (any real or complex number except 0 and negative integers) | dimensionless |
| t | Integration variable | dimensionless |
| e | Euler's number ≈ 2.71828 | dimensionless |
Example 1
Verify that Γ(5) equals 4! = 24 using the recursive property.
Start from Γ(1) = 1 (by definition, since Γ(1) = 0! = 1)
Apply Γ(n+1) = n × Γ(n) repeatedly:
Γ(2) = 1 × Γ(1) = 1 × 1 = 1
Γ(3) = 2 × Γ(2) = 2 × 1 = 2
Γ(4) = 3 × Γ(3) = 3 × 2 = 6
Γ(5) = 4 × Γ(4) = 4 × 6 = 24
Γ(5) = 24, which equals 4! ✓
Example 2
Find Γ(3/2) using the known value Γ(1/2) = √π.
Use the recursive property: Γ(3/2) = Γ(1/2 + 1) = (1/2) × Γ(1/2)
Γ(1/2) = √π ≈ 1.7725
Γ(3/2) = (1/2) × 1.7725 = 0.8862
Γ(3/2) = √π / 2 ≈ 0.8862 — this value appears in the formula for the volume of a 3D sphere
When to Use It
Use the Gamma function when:
- Evaluating factorial-like expressions for non-integer arguments
- Working with probability distributions such as the chi-squared, beta, and gamma distributions
- Normalizing integrals in quantum mechanics and statistical mechanics
- Computing volumes of n-dimensional spheres (hyperspheres)
- Solving differential equations and integrals in advanced physics and engineering