Euler's Gamma Function Γ(x) in 2026 Mathematics
The Gamma function $\Gamma(x)$, introduced by Leonhard Euler in 1729, extends the factorial function from positive integers to all real and complex numbers (except non-positive integers). It satisfies the fundamental functional equation:
\Gamma(x + 1) = x \, \Gamma(x)
For any positive integer $n$:
\Gamma(n) = (n - 1)!
Key Gamma Function Values Reference Table
| Input ($x$) | Exact Gamma Value | Numerical Decimal Approximation | Application |
|---|---|---|---|
| $1$ | $0! = 1$ | 1.0000 | Exponential distribution |
| $2$ | $1! = 1$ | 1.0000 | Baseline |
| $3$ | $2! = 2$ | 2.0000 | Standard |
| $1/2$ | $\sqrt{\pi}$ | 1.77245... | Normal distribution Gaussian integral |
| $3/2$ | $\frac{1}{2}\sqrt{\pi}$ | 0.88622... | Kinetic gas theory |
| $5$ | $4! = 24$ | 24.0000 | Combinatorics |