y = \Gamma(x)

The Gamma Function $y = \Gamma(x)$

The gamma function $\Gamma(x)$ extends the factorial $n!$ beyond the integers to the real numbers, and further to the complex numbers¹. For $x > 0$ it is defined by the following integral.

\Gamma(x) = \int_0^{\infty} t^{x-1} e^{-t}\,dt

Relation to the factorial

Integration by parts yields the recurrence $\Gamma(x+1) = x\,\Gamma(x)$ , and for a positive integer $n$ we have $\Gamma(n) = (n-1)!$ .

$n$	$\Gamma(n)$	$(n-1)!$
$1$	$1$	$0! = 1$
$2$	$1$	$1! = 1$
$3$	$2$	$2! = 2$
$4$	$6$	$3! = 6$
$5$	$24$	$4! = 24$

The offset of $1$ in the index is a convention of the definition; $\Gamma(x+1)$ corresponds to the factorial more directly.

Domain

The recurrence extends the function to $x \leq 0$ , but at $x = 0, -1, -2, \ldots$ the value diverges to infinity. The domain is therefore every real number except those. Writing $\Gamma(x) = \dfrac{\Gamma(x+1)}{x}$ makes the divergence plain: the denominator goes to $0$ as $x \to 0$ .

Asymptotes and limits

There is a vertical asymptote at each of $x = 0, -1, -2, \ldots$ . As $x \to +\infty$ the function grows as sharply as the factorial, and as $x \to 0^{+}$ it diverges to $+\infty$ . On the negative side, narrow spikes reaching to $\pm\infty$ stand between consecutive integers, changing sign from one gap to the next.

Monotonicity and extremum

On $x > 0$ the gamma function is logarithmically convex and attains a minimum at a single point.

Item	Value
$x$ at the minimum	$\approx 1.4616$
$\Gamma(x)$ there	$\approx 0.8856$

That valley lies between the two points where $\Gamma(1) = \Gamma(2) = 1$ . To its left the function decreases and to its right it increases.

Special values and identities

At the half-integers it takes beautiful values.

\Gamma\!\left(\frac{1}{2}\right) = \sqrt{\pi}

This is tied to the Gaussian integral. The reflection formula also holds.

\Gamma(x)\,\Gamma(1-x) = \frac{\pi}{\sin(\pi x)}

Substituting $x = \dfrac{1}{2}$ recovers the value above. That $\sin(\pi x)$ on the right vanishes at the integers corresponds to the gamma function diverging there.

Applications and history

Euler introduced it in the eighteenth century as the problem of interpolating the factorial, and the symbol $\Gamma$ is due to Legendre².

Probability distributions such as the gamma and beta distributions
A basic special function of complex analysis
The continuous version of the binomial coefficient in combinatorics

Gamma function, Wikipedia
Gamma Function, NIST Digital Library of Mathematical Functions

The Gamma Function y=Γ(x)y = \Gamma(x)y=Γ(x)