Gamma Function

Compact study note.

Summary

The gamma function extends factorials from positive integers to positive real and complex arguments. It normalizes gamma, chi-square, t, and F densities.^[1]

Prerequisites

• Prerequisites: None

Notation and Assumptions

For $x>0$ , $\mathrm{\Gamma }(x)={\int }_0^{\mathrm{\infty }}{t}^{x-1}{e}^{-t}dt$ .

Essential Result

$\mathrm{\Gamma }(x+1)=x\mathrm{\Gamma }(x)$ and $\mathrm{\Gamma }(n)=(n-1)!$ for positive integers $n$ .

Small Example

$\mathrm{\Gamma }(3)=2\mathrm{\Gamma }(2)=2\cdot 1\mathrm{\Gamma }(1)=2!$ .

Common Mistakes

• Using $\mathrm{\Gamma }(n)=n!$ instead of $\mathrm{\Gamma }(n)=(n-1)!$ for positive integers.
• Forgetting that distribution formulas often require positive shape parameters.

Connections

• Gamma Distribution
• Chi-Square Distribution
• Student's t Distribution

References