Gamma Function · aiwiki.page

The gamma function, denoted by $\Gamma(z)$ , is a function that extends the factorial through the identity $\Gamma(n+1)=n!$ for nonnegative integers $n$ . It is defined by an integral when the argument has positive real part and extended to almost all complex numbers by analytic continuation. Its integral representation, functional identities, and asymptotic behavior make it a central function in mathematical analysis and probability. (dlmf.nist.gov)

Integral definition and the factorial relation

For $\operatorname{Re}z>0$ , Euler’s integral defines the gamma function:

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

Here $t^{z-1}=\exp((z-1)\ln t)$ , with the ordinary real logarithm because $t>0$ . The condition on $z$ ensures integrability at zero; exponential decay ensures convergence at infinity. For positive real numbers $x$ , the integrand is positive, so $\Gamma(x)>0$ . (dlmf.nist.gov)

Integration by parts gives the recurrence relation

\Gamma(z+1)=z\Gamma(z).

Since $\Gamma(1)=\int_0^\infty e^{-t}\,dt=1$ , induction yields

\Gamma(n+1)=n!, \qquad n=0,1,2,\ldots.

The shift is important: $\Gamma(n)=(n-1)!$ , not $n!$ , for positive integers $n$ . When factorial notation is extended to noninteger arguments, it conventionally means $z!=\Gamma(z+1)$ . (dlmf.nist.gov)

Analytic continuation and singularities

The recurrence extends the function beyond the region where Euler’s integral converges. For an integer $m\geq1$ chosen so that $\operatorname{Re}(z+m)>0$ ,

\Gamma(z)= \frac{\Gamma(z+m)} {z(z+1)\cdots(z+m-1)},

provided the denominator is nonzero. This agrees with the original integral on their common domain and supplies its analytic continuation. (dlmf.nist.gov)

The resulting function is meromorphic: it is analytic except for simple poles at

z=0,-1,-2,\ldots.

At $z=-n$ , the residue is

\operatorname{Res}_{z=-n}\Gamma(z)=\frac{(-1)^n}{n!}.

The gamma function has no zeros. Its reciprocal $1/\Gamma(z)$ , with the singularities removed, is analytic throughout the complex plane and has simple zeros at precisely the nonpositive integers. Thus analytic continuation does not assign finite gamma values at those poles. (dlmf.nist.gov)

Special values and functional identities

The principal noninteger special value is

\Gamma\left(\frac12\right)=\sqrt{\pi}.

Indeed, substituting $t=u^2$ into Euler’s integral gives

\Gamma\left(\frac12\right) =2\int_0^\infty e^{-u^2}\,du =\sqrt{\pi},

using the Gaussian integral. Repeated application of the recurrence then gives

\Gamma\left(n+\frac12\right) =\frac{(2n)!}{4^n n!}\sqrt{\pi}, \qquad n=0,1,2,\ldots.

For example, $\Gamma(3/2)=\sqrt{\pi}/2$ and $\Gamma(-1/2)=-2\sqrt{\pi}$ . (dlmf.nist.gov)

Euler’s reflection formula relates complementary arguments:

\Gamma(z)\Gamma(1-z) =\frac{\pi}{\sin(\pi z)}, \qquad z\notin\mathbb Z.

Legendre’s duplication formula is

\Gamma(z)\Gamma\left(z+\frac12\right) =2^{1-2z}\sqrt{\pi}\,\Gamma(2z).

More generally, Gauss’s multiplication formula, for a positive integer $m$ , states

\prod_{k=0}^{m-1}\Gamma\left(z+\frac{k}{m}\right) =(2\pi)^{(m-1)/2}m^{1/2-mz}\Gamma(mz).

The multiplication identities hold wherever the displayed values are finite, and as identities of meromorphic functions. (dlmf.nist.gov)

Why this extension of factorials is distinguished

Factorial interpolation and the recurrence alone do not determine a unique function. For example, multiplying $\Gamma(x)$ by a positive period-one function $p(x)$ satisfying $p(1)=1$ preserves both the recurrence and the positive-integer values. An additional condition is therefore necessary. (arxiv.org)

The Bohr–Mollerup theorem provides that condition. A positive function $f$ on $(0,\infty)$ equals $\Gamma$ if

$f(1)=1$ ;
$f(x+1)=xf(x)$ ;
$\ln f(x)$ is a convex function.

This logarithmic convexity distinguishes the gamma function from other factorial interpolations. (dlmf.nist.gov)

Logarithmic convexity can also be seen through the second derivative:

\frac{d^2}{dx^2}\ln\Gamma(x) =\sum_{k=0}^\infty\frac{1}{(x+k)^2}>0, \qquad x>0.

It is therefore strict on the positive real axis. The gamma function has a unique positive-real minimum at approximately $x=1.461632$ , where its value is approximately $0.885603$ ; it decreases before this point and increases afterward. (visiblecement.nist.gov)

Product representations

An alternative to the integral definition is Euler’s limit:

\Gamma(z)= \lim_{N\to\infty} \frac{N!\,N^z}{z(z+1)\cdots(z+N)}, \qquad z\notin\{0,-1,-2,\ldots\}.

The Weierstrass product expresses the reciprocal:

\frac{1}{\Gamma(z)} =z e^{\gamma z} \prod_{n=1}^{\infty} \left(1+\frac{z}{n}\right)e^{-z/n},

where $\gamma\approx0.5772156649$ is the Euler–Mascheroni constant, not the gamma function itself. The product displays the reciprocal’s zeros at zero and the negative integers. The exponential factors compensate for the first-order terms of the individual factors, allowing the infinite product to converge. (dlmf.nist.gov)

Asymptotic behavior

For large positive $x$ , Stirling’s approximation gives

\Gamma(x)\sim\sqrt{2\pi}\,x^{x-1/2}e^{-x}.

Here $\sim$ means that the ratio of the two sides tends to one. A more precise asymptotic expansion is

\Gamma(x)\sim \sqrt{2\pi}\,x^{x-1/2}e^{-x} \left( 1+\frac{1}{12x} +\frac{1}{288x^2} -\frac{139}{51840x^3} +\cdots \right).

Equivalently,

\ln\Gamma(x)\sim \left(x-\frac12\right)\ln x-x+\frac12\ln(2\pi) +\frac{1}{12x}-\frac{1}{360x^3} +\frac{1}{1260x^5}-\cdots.

These are asymptotic, generally divergent expansions rather than convergent series at a fixed argument. Truncation is essential to their numerical use. Complex versions hold in sectors bounded away from the negative real axis, with appropriate logarithm conventions. (dlmf.nist.gov)

Ratios have a particularly simple leading behavior. For fixed $a$ and $b$ ,

\frac{\Gamma(z+a)}{\Gamma(z+b)} \sim z^{a-b}

as $|z|\to\infty$ in such sectors. This converts many expressions involving gamma functions into power-law estimates. (dlmf.nist.gov)

Related functions

Beta function

The mathematical beta function is defined, for $\operatorname{Re}a>0$ and $\operatorname{Re}b>0$ , by

B(a,b)=\int_0^1 t^{a-1}(1-t)^{b-1}\,dt.

Its relation to the gamma function is

B(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}.

This identity connects integrals over a finite interval with Euler’s integral over the positive half-line. (dlmf.nist.gov)

Digamma and polygamma functions

The digamma function is the logarithmic derivative

\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}.

Its successive derivatives are the polygamma functions. In particular, $\psi'(z)$ is called the trigamma function. They encode derivatives of expressions containing gamma functions, including the curvature of $\ln\Gamma(x)$ . (dlmf.nist.gov)

Incomplete gamma functions

For real $a>0$ and $x\geq0$ , the lower and upper incomplete gamma functions divide Euler’s integral at $x$ :

\gamma(a,x)=\int_0^x t^{a-1}e^{-t}\,dt, \qquad \Gamma(a,x)=\int_x^\infty t^{a-1}e^{-t}\,dt.

They satisfy

\gamma(a,x)+\Gamma(a,x)=\Gamma(a).

Their normalized forms,

P(a,x)=\frac{\gamma(a,x)}{\Gamma(a)}, \qquad Q(a,x)=\frac{\Gamma(a,x)}{\Gamma(a)},

satisfy $P(a,x)+Q(a,x)=1$ . The two-argument notation $\Gamma(a,x)$ denotes the upper incomplete function, not the ordinary gamma function. (dlmf.nist.gov)

Applications

Probability and statistics

The gamma distribution has probability density

f(x)=\frac{x^{a-1}e^{-x/\theta}} {\Gamma(a)\theta^a}, \qquad x>0,

where $a>0$ is the shape and $\theta>0$ the scale. The denominator normalizes the density. Substitution $t=x/\theta$ reduces its total integral to $\Gamma(a)/\Gamma(a)=1$ . Its cumulative distribution function is $P(a,x/\theta)$ . The chi-squared distribution with $\nu$ degrees of freedom is the case $a=\nu/2$ , $\theta=2$ . (itl.nist.gov)

Geometry and integral representations

In $n$ -dimensional Euclidean space, the volume of a ball of radius $r$ is

V_n(r)=\frac{\pi^{n/2}r^n}{\Gamma(n/2+1)},

and its boundary has surface measure

S_{n-1}(r)= \frac{2\pi^{n/2}r^{n-1}}{\Gamma(n/2)}.

Integer and half-integer gamma values place the formulas for both even and odd dimensions in a single expression. (dlmf.nist.gov)

Gamma functions also form the building blocks of Mellin–Barnes integrals, contour representations of many special functions. Moving the integration contour and evaluating residues can produce expansions for small or large arguments, together with integral representations of the error. (dlmf.nist.gov)

Numerical evaluation and limitations

Methods in numerical analysis combine recurrence, series expansions, asymptotic expansions, and reflection. One approach shifts the argument to a region where an asymptotic expansion is accurate and then applies the recurrence backward. Reflection can transfer evaluation from the left half-plane to the right half-plane. Numerical integration provides another approach. (dlmf.nist.gov)

Because $\Gamma(x)$ grows rapidly, computing $\ln\Gamma(x)$ can avoid overflow in floating-point arithmetic. Gamma ratios and products can often be evaluated through sums and differences of logarithms rather than through separately computed large values. For complex arguments, an analytic logarithm of $\Gamma(z)$ requires branch conventions and need not equal the principal logarithm of a computed gamma value. Near poles, formulas must account for the singular behavior rather than treat the function as finite. (dlmf.nist.gov)

Historical development

The gamma function originated in Leonhard Euler’s study of factorial interpolation in the eighteenth century. Euler developed integral and product descriptions and discovered the reflection formula. Legendre derived the duplication formula, while later work supplied global analytic representations and uniqueness characterizations. The development of these properties transformed factorial interpolation into a broader theory linking analysis, geometry, and statistics. (nist.gov)

References

DLMF: §5.2 Definitionsdlmf.nist.gov
DLMF: §5.4 Special Values and Extremadlmf.nist.gov
DLMF: §5.5 Functional Relationsdlmf.nist.gov
DLMF: §5.8 Infinite Productsdlmf.nist.gov
DLMF: §5.11 Asymptotic Expansionsdlmf.nist.gov
DLMF: §5.12 Beta Functiondlmf.nist.gov
DLMF: §5.15 Polygamma Functionsdlmf.nist.gov
DLMF: §5.19 Mathematical Applicationsdlmf.nist.gov
DLMF: §5.21 Methods of Computationdlmf.nist.gov
DLMF: §8.2 Definitions and Basic Propertiesdlmf.nist.gov
1.6.5. Gammaitl.nist.gov
Gamma and Factorial in the Monthlyarxiv.org