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Gamma function

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Extension of the factorial function

This article uses technical mathematical notation for logarithms. All instances oflog(x) without a subscript base should be interpreted as anatural logarithm, also commonly written asln(x) orlog_e(x).

For the gamma function of ordinals, seeVeblen function. For the gamma distribution in statistics, seeGamma distribution. For the function used in video and image color representations, seeGamma correction.

Gamma
The gamma function along part of the real axis
General information
General definition	$\Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt$
Fields of application	Calculus, mathematical analysis, statistics, physics

Inmathematics, thegamma function (represented by $\Gamma$ , capitalGreek lettergamma) is the most common extension of thefactorial function tocomplex numbers. First studied byDaniel Bernoulli, the gamma function $\Gamma (z)$ is defined for all complex numbers $z {\displaystyle z}$ except non-positive integers, and $\Gamma (n)=(n-1)!$ for everypositive integer $n {\displaystyle n}$ . The gamma function can be defined via a convergentimproper integral for complex numbers with positive real part:

$\Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt,\ \qquad \Re (z)>0.$ The gamma function then is defined in the complex plane as theanalytic continuation of this integral function: it is ameromorphic function which isholomorphic except at zero and the negative integers, where it has simplepoles.

Since the gamma function has no zeros,its reciprocal ${\frac {1}{\Gamma }}$ is anentire function. In fact, the gamma function corresponds to theMellin transform of theexponential decay:

$\Gamma (z)={\mathcal {M}}\{e^{-x}\}(z)\,.$

Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It appears as a factor in various probability-distribution functions and other formulas in the fields ofprobability,statistics,analytic number theory, andcombinatorics.

Motivation

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$\Gamma (x+1)$ interpolates the factorial function to non-integer values.

{\displaystyle \Gamma (x+1)} — $\Gamma (x+1)$ interpolates the factorial function to non-integer values.

The gamma function can be seen as a solution to theinterpolation problem of finding asmooth curve $y=f(x)$ that connects the points of the factorial sequence: $(x,y)=(n,n!)$ for all positive integer values of⁠ $n {\displaystyle n}$ ⁠. The simple formula for the factorial, $x!=1\cdot 2\cdot 3\cdots x$ is only valid when $x {\displaystyle x}$ is a positive integer, and noelementary function has this property, but a good solution is the gamma function⁠ $f(x)=\Gamma (x+1)$ ⁠.^[1]

The gamma function is not only smooth butanalytic (except at the non-positive integers), and it can be defined in several explicit ways. However, it is not the only analytic function that extends the factorial, as one may add any analytic function that is zero on the positive integers, such as $k\sin(m\pi x)$ for an integer⁠ $m {\displaystyle m}$ ⁠.^[1] Such a function is known as apseudogamma function, the most famous being theHadamard function.^[2]

The gamma function,Γ(z) in blue, plotted along withΓ(z) + sin(πz) in green. Notice the intersection at positive integers. Both are valid extensions of the factorials to a meromorphic function on the complex plane.

A more restrictive requirement is thefunctional equation which interpolates the shifted factorial $f(n)=(n{-}1)!$ :^[3]^[4] $f(x+1)=xf(x)\ {\text{ for all }}x>0,\qquad f(1)=1.$

But this still does not give a unique solution, since it allows for multiplication by any periodic function $g(x)$ with $g(x)=g(x+1)$ and⁠ $g(0)=1$ ⁠, such as⁠ $g(x)=e^{k\sin(m\pi x)}$ ⁠.

One way to resolve the ambiguity is theBohr–Mollerup theorem, which shows that $f(x)=\Gamma (x)$ is the unique interpolating function for the factorial, defined over the positive reals, which islogarithmically convex,^[5] meaning that $y=\log f(x)$ isconvex.^[6]

Definition

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Main definition

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The notation $\Gamma (z)$ is due toLegendre.^[1] If the real part of the complex number $z {\displaystyle z}$ is strictly positive ( $\Re (z)>0$ ), then theintegral $\Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt$ converges absolutely, and is known as theEuler integral of the second kind. (Euler's integral of the first kind is thebeta function.^[1])

Absolute value (vertical) and argument (color) of the gamma function on the complex plane

The value $\Gamma (1)$ can be calculated as ${\begin{aligned}\Gamma (1)&=\int _{0}^{\infty }t^{1-1}e^{-t}\,dt\\[6pt]&=\int _{0}^{\infty }e^{-t}\,dt=1.\end{aligned}}$

Integrating by parts, one sees that ${\begin{aligned}\Gamma (z+1)&=\int _{0}^{\infty }t^{z}e^{-t}\,dt\\[6pt]&={\Bigl [}-t^{z}e^{-t}{\Bigr ]}_{0}^{\infty }+\int _{0}^{\infty }z\,t^{z-1}e^{-t}\,dt.\end{aligned}}$ Recognizing that $-t^{z}e^{-t}\to 0$ as⁠ $t\to 0$ ⁠ (so long as $\Re (z)>0$ ) and as⁠ $t\to \infty$ ⁠, ${\begin{aligned}\Gamma (z+1)&=z\int _{0}^{\infty }t^{z-1}e^{-t}\,dt\\[6pt]&=z\,\Gamma (z).\end{aligned}}$

Thus we have shown that $\Gamma (n)=(n-1)!$ for any positive integern byinduction.

The identity ${\textstyle \Gamma (z)={\frac {\Gamma (z+1)}{z}}}$ can be used (or, yielding the same result,analytic continuation can be used) to uniquely extend the integral formulation for $\Gamma (z)$ to ameromorphic function defined for all complex numbers $z {\displaystyle z}$ , except integers less than or equal to zero.^[1] It is this extended version that is commonly referred to as the gamma function.^[1]

Alternative definitions

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There are many equivalent definitions.

Euler's definition as an infinite product

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For a fixed integer⁠ $m {\displaystyle m}$ ⁠, as the integer $n {\displaystyle n}$ increases, we have that^[7] $\lim _{n\to \infty }{\frac {n!\,\left(n+1\right)^{m}}{(n+m)!}}=1\,.$

If $m {\displaystyle m}$ is not an integer then this equation is meaningless since, in this section, the factorial of a non-integer has not been defined yet. However, in order to define the Gamma function for non-integers, let us assume that this equation continues to hold when $m {\displaystyle m}$ is replaced by an arbitrary complex number $z {\displaystyle z}$ :

$\lim _{n\to \infty }{\frac {n!\,\left(n+1\right)^{z}}{(n+z)!}}=1\,.$ Multiplying both sides by $(z-1)!$ gives ${\begin{aligned}(z-1)!&={\frac {1}{z}}\lim _{n\to \infty }n!{\frac {z!}{(n+z)!}}(n+1)^{z}\\[6pt]&={\frac {1}{z}}\lim _{n\to \infty }(1\cdot 2\cdots n){\frac {1}{(1+z)\cdots (n+z)}}\left({\frac {2}{1}}\cdot {\frac {3}{2}}\cdots {\frac {n+1}{n}}\right)^{z}\\[6pt]&={\frac {1}{z}}\prod _{n=1}^{\infty }\left[{\frac {1}{1+{\frac {z}{n}}}}\left(1+{\frac {1}{n}}\right)^{z}\right].\end{aligned}}$ Thisinfinite product, which is due to Euler,^[8] converges for all complex numbers $z {\displaystyle z}$ except the non-positive integers, which fail because of a division by zero. In fact, the above assumption produces a unique definition of $\Gamma (z)$ as⁠ $(z-1)!$ ⁠.

Intuitively, this formula indicates that $\Gamma (z)$ is approximately the result of computing $\Gamma (n+1)=n!$ for some large integer⁠ $n {\displaystyle n}$ ⁠, multiplying by $(n+1)^{z}$ to approximate⁠ $\Gamma (n+z+1)$ ⁠, and then using the relationship $\Gamma (x+1)=x\Gamma (x)$ backwards $n+1$ times to get an approximation for $\Gamma (z)$ ; and furthermore that this approximation becomes exact as $n {\displaystyle n}$ increases to infinity.

The infinite product for thereciprocal ${\frac {1}{\Gamma (z)}}=z\prod _{n=1}^{\infty }\left[\left(1+{\frac {z}{n}}\right)/{\left(1+{\frac {1}{n}}\right)^{z}}\right]$ is anentire function, converging for every complex number $z {\displaystyle z}$ .

Weierstrass's definition

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The definition for the gamma function due toWeierstrass is also valid for all complex numbers $z {\displaystyle z}$ except non-positive integers: $\Gamma (z)={\frac {e^{-\gamma z}}{z}}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)^{-1}e^{z/n},$ where $\gamma \approx 0.577216$ is theEuler–Mascheroni constant.^[1] This is theHadamard product of ${\frac {1}{\Gamma (z)}}$ in a rewritten form.

Proof of equivalence of the three definitions

Equivalence of the integral definition and Weierstrass definition

By the integral definition, the relation $\Gamma (z+1)=z\Gamma (z)$ andHadamard factorization theorem, ${\frac {1}{\Gamma (z)}}=ze^{c_{1}z+c_{2}}\prod _{n=1}^{\infty }e^{-{\frac {z}{n}}}\left(1+{\frac {z}{n}}\right)$ for some constants $c_{1},c_{2}$ since $1/\Gamma$ is an entire function of order⁠ $1 {\displaystyle 1}$ ⁠. Since $z\Gamma (z)\to 1$ as⁠ $z\to 0$ ⁠, $c_{2}=0$ (or an integer multiple of $2\pi i$ ) and since⁠ $\Gamma (1)=1$ ⁠, ${\begin{aligned}e^{-c_{1}}&=\prod _{n=1}^{\infty }e^{-{\frac {1}{n}}}\left(1+{\frac {1}{n}}\right)\\&=\exp \left(\lim _{N\to \infty }\sum _{n=1}^{N}\left(\log \left(1+{\frac {1}{n}}\right)-{\frac {1}{n}}\right)\right)\\&=\exp \left(\lim _{N\to \infty }\left(\log(N+1)-\sum _{n=1}^{N}{\frac {1}{n}}\right)\right).\end{aligned}}$

where $c_{1}=\gamma +2\pi ik$ for some integer⁠ $k {\displaystyle k}$ ⁠. Since $\Gamma (z)\in \mathbb {R}$ for⁠ $z\in \mathbb {R} \setminus \mathbb {Z} _{0}^{-}$ ⁠, we have $k=0$ and ${\frac {1}{\Gamma (z)}}=ze^{\gamma z}\prod _{n=1}^{\infty }e^{-{\frac {z}{n}}}\left(1+{\frac {z}{n}}\right)$

Equivalence of the Weierstrass definition and Euler definition

${\begin{aligned}\Gamma (z)&={\frac {e^{-\gamma z}}{z}}\prod _{n=1}^{\infty }\left(1+{\frac {z}{n}}\right)^{-1}e^{z/n}\\&={\frac {1}{z}}\lim _{n\to \infty }e^{z\left(\log(n+1)-1-{\frac {1}{2}}-{\frac {1}{3}}-\cdots -{\frac {1}{n}}\right)}{\frac {e^{z\left(1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}\right)}}{\left(1+z\right)\left(1+{\frac {z}{2}}\right)\cdots \left(1+{\frac {z}{n}}\right)}}\\&={\frac {1}{z}}\lim _{n\to \infty }{\frac {1}{\left(1+z\right)\left(1+{\frac {z}{2}}\right)\cdots \left(1+{\frac {z}{n}}\right)}}e^{z\log \left(n+1\right)}\\&=\lim _{n\to \infty }{\frac {n!(n+1)^{z}}{z(z+1)\cdots (z+n)}},\quad z\in \mathbb {C} \setminus \mathbb {Z} _{0}^{-}\end{aligned}}$

Properties

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General

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Besides the fundamental property discussed above, $\Gamma (z+1)=z\ \Gamma (z).$ Other important functional equations for the gamma function areEuler's reflection formula, $\Gamma (1-z)\Gamma (z)={\frac {\pi }{\sin \pi z}},\qquad z\not \in \mathbb {Z}$ which implies $\Gamma (z-n)=(-1)^{n-1}\;{\frac {\Gamma (-z)\Gamma (1+z)}{\Gamma (n+1-z)}},\qquad n\in \mathbb {Z}$ and theLegendre duplication formula $\Gamma (z)\Gamma \left(z+{\frac {1}{2}}\right)=2^{1-2z}\,{\sqrt {\pi }}\,\Gamma (2z).$

Derivation of Euler's reflection formula

Proof 1

With Euler's infinite product $\Gamma (z)={\frac {1}{z}}\prod _{n=1}^{\infty }{\frac {(1+1/n)^{z}}{1+z/n}}$ compute ${\frac {1}{\Gamma (1-z)\Gamma (z)}}={\frac {1}{(-z)\Gamma (-z)\Gamma (z)}}=z\prod _{n=1}^{\infty }{\frac {(1-z/n)(1+z/n)}{(1+1/n)^{-z}(1+1/n)^{z}}}=z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right)={\frac {\sin \pi z}{\pi }}\,,$ where the last equality is aknown result. A similar derivation begins with Weierstrass's definition.

Proof 2

First prove that $I=\int _{-\infty }^{\infty }{\frac {e^{ax}}{1+e^{x}}}\,dx=\int _{0}^{\infty }{\frac {v^{a-1}}{1+v}}\,dv={\frac {\pi }{\sin \pi a}},\quad a\in (0,1).$ Consider the positively oriented rectangular contour $C_{R}$ with vertices at⁠ $R {\displaystyle R}$ ⁠,⁠ $-R$ ⁠, $R+2\pi i$ and $-R+2\pi i$ where⁠ $R\in \mathbb {R} ^{+}$ ⁠. Then by theresidue theorem, $\int _{C_{R}}{\frac {e^{az}}{1+e^{z}}}\,dz=-2\pi ie^{a\pi i}.$ Let $I_{R}=\int _{-R}^{R}{\frac {e^{ax}}{1+e^{x}}}\,dx$ and let $I_{R}'$ be the analogous integral over the top side of the rectangle. Then $I_{R}\to I$ as $R\to \infty$ and⁠ $I_{R}'=-e^{2\pi ia}I_{R}$ ⁠. If $A_{R}$ denotes the right vertical side of the rectangle, then $\left|\int _{A_{R}}{\frac {e^{az}}{1+e^{z}}}\,dz\right|\leq \int _{0}^{2\pi }\left|{\frac {e^{a(R+it)}}{1+e^{R+it}}}\right|\,dt\leq Ce^{(a-1)R}$ for some constant $C {\displaystyle C}$ and since⁠ $a<1$ ⁠, the integral tends to $0 {\displaystyle 0}$ as⁠ $R\to \infty$ ⁠. Analogously, the integral over the left vertical side of the rectangle tends to $0 {\displaystyle 0}$ as⁠ $R\to \infty$ ⁠. Therefore $I-e^{2\pi ia}I=-2\pi ie^{a\pi i},$ from which $I={\frac {\pi }{\sin \pi a}},\quad a\in (0,1).$ Then $\Gamma (1-z)=\int _{0}^{\infty }e^{-u}u^{-z}\,du=t\int _{0}^{\infty }e^{-vt}(vt)^{-z}\,dv,\quad t>0$ and ${\begin{aligned}\Gamma (z)\Gamma (1-z)&=\int _{0}^{\infty }\int _{0}^{\infty }e^{-t(1+v)}v^{-z}\,dv\,dt\\&=\int _{0}^{\infty }{\frac {v^{-z}}{1+v}}\,dv\\&={\frac {\pi }{\sin \pi (1-z)}}\\&={\frac {\pi }{\sin \pi z}},\quad z\in (0,1).\end{aligned}}$ Proving the reflection formula for all $z\in (0,1)$ proves it for all $z\in \mathbb {C} \setminus \mathbb {Z}$ by analytic continuation.

Derivation of the Legendre duplication formula

Thebeta function can be represented as $\mathrm {B} (z_{1},z_{2})={\frac {\Gamma (z_{1})\Gamma (z_{2})}{\Gamma (z_{1}+z_{2})}}=\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt.$

Setting $z_{1}=z_{2}=z$ yields ${\frac {\Gamma ^{2}(z)}{\Gamma (2z)}}=\int _{0}^{1}t^{z-1}(1-t)^{z-1}\,dt.$

After the substitution $t={\frac {1+u}{2}}$ : ${\frac {\Gamma ^{2}(z)}{\Gamma (2z)}}={\frac {1}{2^{2z-1}}}\int _{-1}^{1}\left(1-u^{2}\right)^{z-1}\,du.$

The function $(1-u^{2})^{z-1}$ is even, hence $2^{2z-1}\Gamma ^{2}(z)=2\Gamma (2z)\int _{0}^{1}(1-u^{2})^{z-1}\,du.$

Now $\mathrm {B} \left({\frac {1}{2}},z\right)=\int _{0}^{1}t^{{\frac {1}{2}}-1}(1-t)^{z-1}\,dt,\quad t=s^{2}.$

Then $\mathrm {B} \left({\frac {1}{2}},z\right)=2\int _{0}^{1}(1-s^{2})^{z-1}\,ds=2\int _{0}^{1}(1-u^{2})^{z-1}\,du.$

This implies $2^{2z-1}\Gamma ^{2}(z)=\Gamma (2z)\mathrm {B} \left({\frac {1}{2}},z\right).$

Since $\mathrm {B} \left({\frac {1}{2}},z\right)={\frac {\Gamma \left({\frac {1}{2}}\right)\Gamma (z)}{\Gamma \left(z+{\frac {1}{2}}\right)}},\quad \Gamma \left({\frac {1}{2}}\right)={\sqrt {\pi }},$ the Legendre duplication formula follows: $\Gamma (z)\Gamma \left(z+{\frac {1}{2}}\right)=2^{1-2z}{\sqrt {\pi }}\;\Gamma (2z).$

The duplication formula is a special case of themultiplication theorem (see ^[9] Eq. 5.5.6): $\prod _{k=0}^{m-1}\Gamma \left(z+{\frac {k}{m}}\right)=(2\pi )^{\frac {m-1}{2}}\;m^{{\frac {1}{2}}-mz}\;\Gamma (mz).$

A simple but useful property, which can be seen from the limit definition, is: ${\overline {\Gamma (z)}}=\Gamma ({\overline {z}})\;\Rightarrow \;\Gamma (z)\Gamma ({\overline {z}})\in \mathbb {R} .$

In particular, withz =a +bi, this product is $|\Gamma (a+bi)|^{2}=|\Gamma (a)|^{2}\prod _{k=0}^{\infty }{\frac {1}{1+{\frac {b^{2}}{(a+k)^{2}}}}}$

If the real part is an integer or ahalf-integer, this can be finitely expressed inclosed form: ${\begin{aligned}|\Gamma (bi)|^{2}&={\frac {\pi }{b\sinh \pi b}}\\[6pt]\left|\Gamma \left({\tfrac {1}{2}}+bi\right)\right|^{2}&={\frac {\pi }{\cosh \pi b}}\\[6pt]\left|\Gamma \left(1+bi\right)\right|^{2}&={\frac {\pi b}{\sinh \pi b}}\\[6pt]\left|\Gamma \left(1+n+bi\right)\right|^{2}&={\frac {\pi b}{\sinh \pi b}}\prod _{k=1}^{n}\left(k^{2}+b^{2}\right),\quad n\in \mathbb {N} \\[6pt]\left|\Gamma \left(-n+bi\right)\right|^{2}&={\frac {\pi }{b\sinh \pi b}}\prod _{k=1}^{n}\left(k^{2}+b^{2}\right)^{-1},\quad n\in \mathbb {N} \\[6pt]\left|\Gamma \left({\tfrac {1}{2}}\pm n+bi\right)\right|^{2}&={\frac {\pi }{\cosh \pi b}}\prod _{k=1}^{n}\left(\left(k-{\tfrac {1}{2}}\right)^{2}+b^{2}\right)^{\pm 1},\quad n\in \mathbb {N} \\[-1ex]&\end{aligned}}$

Proof of absolute value formulas for arguments of integer or half-integer real part

First, consider the reflection formula applied to⁠ $z=bi$ ⁠. $\Gamma (bi)\Gamma (1-bi)={\frac {\pi }{\sin \pi bi}}$ Applying the recurrence relation to the second term: $-bi\cdot \Gamma (bi)\Gamma (-bi)={\frac {\pi }{\sin \pi bi}}$ which with simple rearrangement gives $\Gamma (bi)\Gamma (-bi)={\frac {\pi }{-bi\sin \pi bi}}={\frac {\pi }{b\sinh \pi b}}$

Second, consider the reflection formula applied to⁠ $z={\tfrac {1}{2}}+bi$ ⁠. $\Gamma ({\tfrac {1}{2}}+bi)\Gamma \left(1-({\tfrac {1}{2}}+bi)\right)=\Gamma ({\tfrac {1}{2}}+bi)\Gamma ({\tfrac {1}{2}}-bi)={\frac {\pi }{\sin \pi ({\tfrac {1}{2}}+bi)}}={\frac {\pi }{\cos \pi bi}}={\frac {\pi }{\cosh \pi b}}$

Formulas for other values of $z {\displaystyle z}$ for which the real part is integer or half-integer quickly follow byinduction using the recurrence relation in the positive and negative directions.

Perhaps the best-known value of the gamma function at a non-integer argument is $\Gamma \left({\tfrac {1}{2}}\right)={\sqrt {\pi }},$ which can be found by setting ${\textstyle z={\frac {1}{2}}}$ in the reflection formula, by using the relation to thebeta function given below with⁠ $z_{1}=z_{2}={\frac {1}{2}}$ ⁠, or simply by making the substitution $t=u^{2}$ in the integral definition of the gamma function, resulting in aGaussian integral. In general, for non-negative integer values of $n {\displaystyle n}$ we have: ${\begin{aligned}\Gamma \left({\frac {1}{2}}+n\right)&={(2n)! \over 4^{n}n!}{\sqrt {\pi }}={\frac {(2n-1)!!}{2^{n}}}{\sqrt {\pi }}={\binom {n-{\frac {1}{2}}}{n}}\,n!\,{\sqrt {\pi }}\\[6pt]\Gamma \left({\frac {1}{2}}-n\right)&={(-4)^{n}n! \over (2n)!}{\sqrt {\pi }}={\frac {(-2)^{n}}{(2n-1)!!}}{\sqrt {\pi }}={\frac {\sqrt {\pi }}{{\binom {-1/2}{n}}\,n!}}\end{aligned}}$ where thedouble factorial⁠ $(2n-1)!!=(2n-1)(2n-3)\cdots (3)(1)$ ⁠. SeeParticular values of the gamma function for calculated values.

It might be tempting to generalize the result that ${\textstyle \Gamma \left({\frac {1}{2}}\right)={\sqrt {\pi }}}$ by looking for a formula for other individual values $\Gamma (r)$ where $r {\displaystyle r}$ is rational, especially because according toGauss's digamma theorem, it is possible to do so for the closely relateddigamma function at every rational value. However, these numbers $\Gamma (r)$ are not known to be expressible by themselves in terms of elementary functions. It has been proved that $\Gamma (n+r)$ is atranscendental number andalgebraically independent of $\pi$ for any integer $n {\displaystyle n}$ and each of the fractions ${\textstyle r={\frac {1}{6}},{\frac {1}{4}},{\frac {1}{3}},{\frac {2}{3}},{\frac {3}{4}},{\frac {5}{6}}}$ .^[10] In general, when computing values of the gamma function, we must settle for numerical approximations.

The derivatives of the gamma function are described in terms of thepolygamma function, $\psi ^{(m)}(z)$ : $\Gamma '(z)=\Gamma (z)\psi ^{(0)}(z).$ For a positive integer $m {\displaystyle m}$ the derivative of the gamma function can be calculated as follows:

Gamma function in the complex plane with colors showing its argument — Colors showing the argument of the gamma function in the complex plane from−2 − 2i to6 + 2i

$\Gamma '(m+1)=m!\left(-\gamma +\sum _{k=1}^{m}{\frac {1}{k}}\right)=m!\left(-\gamma +H(m)\right)\,,$ where $H(m)$ is the $m {\displaystyle m}$ thharmonic number and $\gamma$ is theEuler–Mascheroni constant.

For $\Re (z)>0$ the $n {\displaystyle n}$ th derivative of the gamma function is: ${\frac {d^{n}}{dz^{n}}}\Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}(\log t)^{n}\,dt.$ (This can be derived bydifferentiating the integral form of the gamma function with respect to⁠ $z {\displaystyle z}$ ⁠.)

Using the identity $\Gamma ^{(n)}(1)=(-1)^{n}B_{n}(\gamma ,1!\zeta (2),\ldots ,(n-1)!\,\zeta (n)),$ where $\zeta (z)$ is theRiemann zeta function, and $B_{n}$ is the $n {\displaystyle n}$ -thBell polynomial, we have in particular theLaurent series expansion of the gamma function^[11] $\Gamma (z)={\frac {1}{z}}-\gamma +{\frac {1}{2}}\left(\gamma ^{2}+{\frac {\pi ^{2}}{6}}\right)z-{\frac {1}{6}}\left(\gamma ^{3}+{\frac {\gamma \pi ^{2}}{2}}+2\zeta (3)\right)z^{2}+O(z^{3}).$

Inequalities

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When restricted to thepositive real numbers, the gamma function is a strictlylogarithmically convex function. This property may be stated in any of the following three equivalent ways:

For any two positive real numbers $x_{1}$ and $x_{2}$ , and for any $t\in [0,1]$ , $\Gamma (tx_{1}+(1-t)x_{2})\leq \Gamma (x_{1})^{t}\Gamma (x_{2})^{1-t}.$
For any two positive real numbers $x_{1}$ and $x_{2}$ , and $x_{2}$ > $x_{1}$ $\left({\frac {\Gamma (x_{2})}{\Gamma (x_{1})}}\right)^{\frac {1}{x_{2}-x_{1}}}>\exp \left({\frac {\Gamma '(x_{1})}{\Gamma (x_{1})}}\right).$
For any positive real number⁠ $x {\displaystyle x}$ ⁠, $\Gamma ''(x)\Gamma (x)>\Gamma '(x)^{2}.$

The last of these statements is, essentially by definition, the same as the statement that $\psi ^{(1)}(x)>0$ , where $\psi ^{(1)}$ is thepolygamma function of order 1. To prove the logarithmic convexity of the gamma function, it therefore suffices to observe that $\psi ^{(1)}$ has a series representation which, for positive realx, consists of only positive terms.

Logarithmic convexity andJensen's inequality together imply, for any positive real numbers $x_{1},\ldots ,x_{n}$ and $a_{1},\ldots ,a_{n}$ , $\Gamma \left({\frac {a_{1}x_{1}+\cdots +a_{n}x_{n}}{a_{1}+\cdots +a_{n}}}\right)\leq {\bigl (}\Gamma (x_{1})^{a_{1}}\cdots \Gamma (x_{n})^{a_{n}}{\bigr )}^{\frac {1}{a_{1}+\cdots +a_{n}}}.$

There are also bounds on ratios of gamma functions. The best-known isGautschi's inequality, which says that for any positive real numberx and anys ∈ (0, 1), $x^{1-s}<{\frac {\Gamma (x+1)}{\Gamma (x+s)}}<\left(x+1\right)^{1-s}.$

The behavior for non-positive $z {\displaystyle z}$ is more intricate. Euler's integral does not converge for $\Re (z)\leq 0$ , but the function it defines in the positive complex half-plane has a uniqueanalytic continuation to the negative half-plane. One way to find that analytic continuation is to use Euler's integral for positive arguments and extend the domain to negative numbers by repeated application of the recurrence formula,^[1] $\Gamma (z)={\frac {\Gamma (z+n+1)}{z(z+1)\cdots (z+n)}},$ choosing $n {\displaystyle n}$ such that $z+n$ is positive. The product in the denominator is zero when $z {\displaystyle z}$ equals any of the integers $0,-1,-2,\ldots$ . Thus, the gamma function must be undefined at those points to avoiddivision by zero; it is ameromorphic function withsimple poles at the non-positive integers.^[1]

For a function $f {\displaystyle f}$ of a complex variable⁠ $z {\displaystyle z}$ ⁠, at asimple pole⁠ $c {\displaystyle c}$ ⁠, theresidue of $f {\displaystyle f}$ is given by: $\operatorname {Res} (f,c)=\lim _{z\to c}(z-c)f(z).$

For the simple pole $z=-n$ , the recurrence formula can be rewritten as: $(z+n)\Gamma (z)={\frac {\Gamma (z+n+1)}{z(z+1)\cdots (z+n-1)}}.$ The numerator at⁠ $z=-n$ ⁠, is $\Gamma (z+n+1)=\Gamma (1)=1$ and the denominator $z(z+1)\cdots (z+n-1)=-n(1-n)\cdots (n-1-n)=(-1)^{n}n!.$ So the residues of the gamma function at those points are:^[14] $\operatorname {Res} (\Gamma ,-n)={\frac {(-1)^{n}}{n!}}.$ The gamma function is non-zero everywhere along the real line, although it comes arbitrarily close to zero as $z\rightarrow -\infty$ . There is in fact no complex number $z {\displaystyle z}$ for which $\Gamma (z)=0$ , and hence thereciprocal gamma function ${\textstyle {\dfrac {1}{\Gamma (z)}}}$ is anentire function, with zeros at $z=0,-1,-2,\ldots$ .^[1]

Minima and maxima

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On the real line, the gamma function has a local minimum atz_min ≈+1.46163214496836234126^[15] where it attains the valueΓ(z_min) ≈+0.88560319441088870027.^[16] The gamma function rises to either side of this minimum. The solution toΓ(z − 0.5) = Γ(z + 0.5) isz = +1.5 and the common value isΓ(1) = Γ(2) = +1. The positive solution toΓ(z − 1) = Γ(z + 1) isz =φ ≈ +1.618, thegolden ratio, and the common value isΓ(φ − 1) = Γ(φ + 1) =φ! ≈+1.44922960226989660037.^[17]

The gamma function must alternate sign between its poles at the non-positive integers because the product in the forward recurrence contains an odd number of negative factors if the number of poles between $z {\displaystyle z}$ and $z+n$ is odd, and an even number if the number of poles is even.^[14] The values at the local extrema of the gamma function along the real axis between the non-positive integers are:

Γ(−0.50408300826445540925...^[18]) =−3.54464361115500508912...,

Γ(−1.57349847316239045877...^[19]) =2.30240725833968013582...,

Γ(−2.61072086844414465000...^[20]) =−0.88813635840124192009...,

Γ(−3.63529336643690109783...^[21]) =0.24512753983436625043...,

Γ(−4.65323776174314244171...^[22]) =−0.05277963958731940076..., etc.

Integral representations

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There are many formulas, besides the Euler integral of the second kind, that express the gamma function as an integral. For instance, when the real part of $z {\displaystyle z}$ is positive,^[23] $\Gamma (z)=\int _{-\infty }^{\infty }e^{zt-e^{t}}\,dt$ and^[24] $\Gamma (z)=\int _{0}^{1}\left(\log {\frac {1}{t}}\right)^{z-1}\,dt,$ $\Gamma (z)=2c^{z}\int _{0}^{\infty }t^{2z-1}e^{-ct^{2}}\,dt\,,\;c>0$ where the three integrals respectively follow from the substitutions $t=e^{-x}$ , $t=-\log x$ ^[25] and $t=cx^{2}$ ^[26] in Euler's second integral. The last integral in particular makes clear the connection between the gamma function at half integer arguments and theGaussian integral: if $z=1/2,\;c=1$ we get $\Gamma (1/2)=2\int _{0}^{\infty }e^{-t^{2}}\,dt={\sqrt {\pi }}\;.$

Binet's first integral formula for the gamma function states that, when the real part ofz is positive, then:^[27] $\operatorname {log\Gamma } (z)=\left(z-{\frac {1}{2}}\right)\log z-z+{\frac {1}{2}}\log(2\pi )+\int _{0}^{\infty }\left({\frac {1}{2}}-{\frac {1}{t}}+{\frac {1}{e^{t}-1}}\right){\frac {e^{-tz}}{t}}\,dt.$ The integral on the right-hand side may be interpreted as aLaplace transform. That is, $\log \left(\Gamma (z)\left({\frac {e}{z}}\right)^{z}{\sqrt {\frac {z}{2\pi }}}\right)={\mathcal {L}}\left({\frac {1}{2t}}-{\frac {1}{t^{2}}}+{\frac {1}{t(e^{t}-1)}}\right)(z).$

Binet's second integral formula states that, again when the real part ofz is positive, then:^[28] $\operatorname {log\Gamma } (z)=\left(z-{\frac {1}{2}}\right)\log z-z+{\frac {1}{2}}\log(2\pi )+2\int _{0}^{\infty }{\frac {\arctan(t/z)}{e^{2\pi t}-1}}\,dt.$

LetC be aHankel contour, meaning a path that begins and ends at the point∞ on theRiemann sphere, whose unit tangent vector converges to−1 at the start of the path and to1 at the end, which haswinding number 1 around0, and which does not cross $[0,\infty )$ . Fix a branch of $\log(-t)$ by taking a branch cut along $[0,\infty )$ and by taking $\log(-t)$ to be real when $t {\displaystyle t}$ is on the negative real axis. If $z {\displaystyle z}$ is not an integer, then Hankel's formula for the gamma function is:^[29] $\Gamma (z)=-{\frac {1}{2i\sin \pi z}}\int _{C}(-t)^{z-1}e^{-t}\,dt,$ where $(-t)^{z-1}$ is interpreted as $\exp((z-1)\log(-t))$ . The reflection formula leads to the closely related expression ${\frac {1}{\Gamma (z)}}={\frac {i}{2\pi }}\int _{C}(-t)^{-z}e^{-t}\,dt,$ which is valid whenever $z\not \in \mathbb {Z}$ .

Continued fraction representation

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The gamma function can also be represented by a sum of twocontinued fractions:^[30]^[31] ${\begin{aligned}\Gamma (z)&={\cfrac {e^{-1}}{2+0-z+1{\cfrac {z-1}{2+2-z+2{\cfrac {z-2}{2+4-z+3{\cfrac {z-3}{2+6-z+4{\cfrac {z-4}{2+8-z+5{\cfrac {z-5}{2+10-z+\ddots }}}}}}}}}}}}\\&+\ {\cfrac {e^{-1}}{z+0-{\cfrac {z+0}{z+1+{\cfrac {1}{z+2-{\cfrac {z+1}{z+3+{\cfrac {2}{z+4-{\cfrac {z+2}{z+5+{\cfrac {3}{z+6-\ddots }}}}}}}}}}}}}}\end{aligned}}$ where $z\in \mathbb {C}$ .

Fourier series expansion

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Thelogarithm of the gamma function has the followingFourier series expansion for $0<z<1:$ $\operatorname {log\Gamma } (z)=\left({\frac {1}{2}}-z\right)(\gamma +\log 2)+(1-z)\log \pi -{\frac {1}{2}}\log \sin(\pi z)+{\frac {1}{\pi }}\sum _{n=1}^{\infty }{\frac {\log n}{n}}\sin(2\pi nz),$ which was for a long time attributed toErnst Kummer, who derived it in 1847.^[32]^[33] However, Iaroslav Blagouchine discovered thatCarl Johan Malmsten first derived this series in 1842.^[34]^[35]

Raabe's formula

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In 1840Joseph Ludwig Raabe proved that $\int _{a}^{a+1}\log \Gamma (z)\,dz={\tfrac {1}{2}}\log 2\pi +a\log a-a,\quad a>0.$ In particular, if $a=0$ then $\int _{0}^{1}\log \Gamma (z)\,dz={\frac {1}{2}}\log(2\pi ).$

The latter can be derived taking the logarithm in the above multiplication formula, which gives an expression for the Riemann sum of the integrand. Taking the limit for $a\to \infty$ gives the formula.

Pi function

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An alternative notation introduced byGauss is the $\Pi$ -function, a shifted version of the gamma function: $\Pi (z)=\Gamma (z+1)=z\Gamma (z)=\int _{0}^{\infty }e^{-t}t^{z}\,dt,$ so that $\Pi (n)=n!$ for every non-negative integer⁠ $n {\displaystyle n}$ ⁠.

Using the pi function, the reflection formula is: $\Pi (z)\Pi (-z)={\frac {\pi z}{\sin(\pi z)}}={\frac {1}{\operatorname {sinc} (z)}}$ using the normalizedsinc function; while the multiplication theorem becomes: $\Pi \left({\frac {z}{m}}\right)\,\Pi \left({\frac {z-1}{m}}\right)\cdots \Pi \left({\frac {z-m+1}{m}}\right)=(2\pi )^{\frac {m-1}{2}}m^{-z-{\frac {1}{2}}}\Pi (z)\ .$

The shiftedreciprocal gamma function is sometimes denoted⁠ $\pi (z)={\frac {1}{\Pi (z)}}$ ⁠, anentire function.

Thevolume of ann-ellipsoid with radiir₁, ...,r_n can be expressed as $V_{n}(r_{1},\dotsc ,r_{n})={\frac {\pi ^{\frac {n}{2}}}{\Pi \left({\frac {n}{2}}\right)}}\prod _{k=1}^{n}r_{k}.$

Relation to other functions

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In the first integral defining the gamma function, the limits of integration are fixed. The upperincomplete gamma function is obtained by allowing the lower limit of integration to vary: $\Gamma (z,x)=\int _{x}^{\infty }t^{z-1}e^{-t}dt.$ There is a similar lower incomplete gamma function.
The gamma function is related to Euler'sbeta function by the formula $\mathrm {B} (z_{1},z_{2})=\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt={\frac {\Gamma (z_{1})\,\Gamma (z_{2})}{\Gamma (z_{1}+z_{2})}}.$
Thelogarithmic derivative of the gamma function is called thedigamma function; higher derivatives are thepolygamma functions.
The analog of the gamma function over afinite field or afinite ring is theGaussian sums, a type ofexponential sum.
Thereciprocal gamma function is anentire function and has been studied as a specific topic.
The gamma function also shows up in an important relation with theRiemann zeta function, $\zeta (z)$ . $\pi ^{-{\frac {z}{2}}}\;\Gamma \left({\frac {z}{2}}\right)\zeta (z)=\pi ^{-{\frac {1-z}{2}}}\;\Gamma \left({\frac {1-z}{2}}\right)\;\zeta (1-z).$ It also appears in the following formula: $\zeta (z)\Gamma (z)=\int _{0}^{\infty }{\frac {u^{z}}{e^{u}-1}}\,{\frac {du}{u}},$ which is valid only for $\Re (z)>1$ .
The logarithm of the gamma function satisfies the following formula due to Lerch: $\operatorname {log\Gamma } (z)=\zeta _{H}'(0,z)-\zeta '(0),$ where $\zeta _{H}$ is theHurwitz zeta function, $\zeta$ is the Riemann zeta function and the prime (′) denotes differentiation in the first variable.
The gamma function is related to thestretched exponential function. For instance, the moments of that function are $\langle \tau ^{n}\rangle \equiv \int _{0}^{\infty }t^{n-1}\,e^{-\left({\frac {t}{\tau }}\right)^{\beta }}\,\mathrm {d} t={\frac {\tau ^{n}}{\beta }}\Gamma \left({n \over \beta }\right).$

Particular values

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Main article:Particular values of the gamma function

Including up to the first 20 digits after the decimal point, some particular values of the gamma function are: ${\begin{array}{rcccl}\Gamma \left(-{\frac {3}{2}}\right)&=&{\frac {4{\sqrt {\pi }}}{3}}&\approx &+2.36327\,18012\,07354\,70306\\[6pt]\Gamma \left(-{\frac {1}{2}}\right)&=&-2{\sqrt {\pi }}&\approx &-3.54490\,77018\,11032\,05459\\[6pt]\Gamma \left({\frac {1}{2}}\right)&=&{\sqrt {\pi }}&\approx &+1.77245\,38509\,05516\,02729\\[6pt]\Gamma (1)&=&0!&=&+1\\[6pt]\Gamma \left({\frac {3}{2}}\right)&=&{\frac {\sqrt {\pi }}{2}}&\approx &+0.88622\,69254\,52758\,01364\\[6pt]\Gamma (2)&=&1!&=&+1\\[6pt]\Gamma \left({\frac {5}{2}}\right)&=&{\frac {3{\sqrt {\pi }}}{4}}&\approx &+1.32934\,03881\,79137\,02047\\[6pt]\Gamma (3)&=&2!&=&+2\\[6pt]\Gamma \left({\frac {7}{2}}\right)&=&{\tfrac {15{\sqrt {\pi }}}{8}}&\approx &+3.32335\,09704\,47842\,55118\\[6pt]\Gamma (4)&=&3!&=&+6\end{array}}$ (These numbers can be found in theOEIS.^[36]^[37]^[38]^[39]^[40]^[41] The values presented here are truncated rather than rounded.) The complex-valued gamma function is undefined for non-positive integers, but in these cases the value can be defined in theRiemann sphere as $\infty$ . Thereciprocal gamma function iswell defined andanalytic at these values (and in theentire complex plane): ${\frac {1}{\Gamma (-3)}}={\frac {1}{\Gamma (-2)}}={\frac {1}{\Gamma (-1)}}={\frac {1}{\Gamma (0)}}=0.$

Log-gamma function

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Because the gamma and factorial functions grow so rapidly for moderately large arguments, many computing environments include a function that returns thenatural logarithm of the gamma function, often given the namelgamma orlngamma in programming environments orgammaln in spreadsheets. This grows much more slowly, and for combinatorial calculations allows adding and subtracting logarithmic values instead of multiplying and dividing very large values. It is often defined as^[42] $\operatorname {log\Gamma } (z)=-\gamma z-\log z+\sum _{k=1}^{\infty }\left[{\frac {z}{k}}-\log \left(1+{\frac {z}{k}}\right)\right].$

Thedigamma function, which is the derivative of this function, is also commonly seen.In the context of technical and physical applications, e.g. with wave propagation, the functional equation $\operatorname {log\Gamma } (z)=\operatorname {log\Gamma } (z+1)-\log z$

Logarithmic gamma function in the complex plane from −2 − 2i to 2 + 2i with colors — Logarithmic gamma function in the complex plane
from−2 − 2i to2 + 2i with colors

is often used since it allows one to determine function values in one strip of width 1 in $z {\displaystyle z}$ from the neighbouring strip. In particular, starting with a good approximation for a $z {\displaystyle z}$ with large real part one may go step by step down to the desired $z {\displaystyle z}$ . Following an indication ofCarl Friedrich Gauss, Rocktaeschel (1922) proposed for $\log \Gamma (z)$ an approximation for large $\Re (z)$ : $\operatorname {log\Gamma } (z)\approx (z-{\tfrac {1}{2}})\log z-z+{\tfrac {1}{2}}\log(2\pi ).$

This can be used to accurately approximate $\log \Gamma (z)$ for $z {\displaystyle z}$ with a smaller $\Re (z)$ via (P.E.Böhmer, 1939) $\operatorname {log\Gamma } (z-m)=\operatorname {log\Gamma } (z)-\sum _{k=1}^{m}\log(z-k).$

A more accurate approximation can be obtained by using more terms from the asymptotic expansions of $\log \Gamma (z)$ and $\Gamma (z)$ , which are based on Stirling's approximation. $\Gamma (z)\sim z^{z-{\frac {1}{2}}}e^{-z}{\sqrt {2\pi }}\left(1+{\frac {1}{12z}}+{\frac {1}{288z^{2}}}-{\frac {139}{51\,840z^{3}}}-{\frac {571}{2\,488\,320z^{4}}}\right)$

|z|\rightarrow \infty

at constant

|\arg(z)|<\pi

. (See sequencesA001163 andA001164 in theOEIS.)

In a more "natural" presentation, $\log \Gamma (z)=z\log z-z-{\frac {1}{2}}\log z+{\frac {1}{2}}\log 2\pi +{\frac {1}{12z}}-{\frac {1}{360z^{3}}}+{\frac {1}{1260z^{5}}}+O\left({\frac {1}{z^{5}}}\right)$

|z|\rightarrow \infty

at constant

|\arg(z)|<\pi

. (See sequencesA046968 andA046969 in theOEIS.)

The coefficients of the terms with $k>1$ of $z^{1-k}$ in the last expansion are simply ${\frac {B_{k}}{k(k-1)}},$ where the $B_{k}$ are theBernoulli numbers.

The gamma function also has Stirling Series (derived byCharles Hermite in 1900) equal to^[43] $\operatorname {log\Gamma } (1+x)={\frac {x(x-1)}{2!}}\log(2)+{\frac {x(x-1)(x-2)}{3!}}(\log(3)-2\log(2))+\cdots ,\quad \Re (x)>0.$

Properties

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TheBohr–Mollerup theorem states that among all functions extending the factorial functions to the positive real numbers, only the gamma function islog-convex, that is, itsnatural logarithm isconvex on the positive real axis. Another characterisation is given by theWielandt theorem.

The gamma function is the unique function that simultaneously satisfies

$\Gamma (1)=1$ ,
$\Gamma (z+1)=z\Gamma (z)$ for all complex numbers $z {\displaystyle z}$ except the non-positive integers, and,
for integern, ${\textstyle \lim _{n\to \infty }{\frac {\Gamma (n+z)}{\Gamma (n)\;n^{z}}}=1}$ for all complex numbers⁠ $z {\displaystyle z}$ ⁠.^[1]

In a certain sense, the log-gamma function is the more natural form; it makes some intrinsic attributes of the function clearer. A striking example is theTaylor series oflogΓ around 1: $\operatorname {log\Gamma } (z+1)=-\gamma z+\sum _{k=2}^{\infty }{\frac {\zeta (k)}{k}}\,(-z)^{k}\qquad \forall \;|z|<1$ withζ(k) denoting theRiemann zeta function atk.

So, using the following property: $\zeta (s)\Gamma (s)=\int _{0}^{\infty }{\frac {t^{s}}{e^{t}-1}}\,{\frac {dt}{t}}$ an integral representation for the log-gamma function is: $\operatorname {log\Gamma } (z+1)=-\gamma z+\int _{0}^{\infty }{\frac {e^{-zt}-1+zt}{t\left(e^{t}-1\right)}}\,dt$ or, settingz = 1 to obtain an integral forγ, we can replace theγ term with its integral and incorporate that into the above formula, to get: $\operatorname {log\Gamma } (z+1)=\int _{0}^{\infty }{\frac {e^{-zt}-ze^{-t}-1+z}{t\left(e^{t}-1\right)}}\,dt\,.$

There also exist special formulas for the logarithm of the gamma function for rationalz.For instance, if $k {\displaystyle k}$ and $n {\displaystyle n}$ are integers with $k<n$ and⁠ $k\neq n/2$ ⁠, then^[44] ${\begin{aligned}\operatorname {log\Gamma } \left({\frac {k}{n}}\right)={}&{\frac {\,(n-2k)\log 2\pi \,}{2n}}+{\frac {1}{2}}\left\{\,\log \pi -\log \sin {\frac {\pi k}{n}}\,\right\}+{\frac {1}{\pi }}\!\sum _{r=1}^{n-1}{\frac {\,\gamma +\log r\,}{r}}\cdot \sin {\frac {\,2\pi rk\,}{n}}\\&{}-{\frac {1}{2\pi }}\sin {\frac {2\pi k}{n}}\cdot \!\int _{0}^{\infty }\!\!{\frac {\,e^{-nx}\!\cdot \log x\,}{\,\cosh x-\cos(2\pi k/n)\,}}\,{\mathrm {d} }x.\end{aligned}}$ This formula is sometimes used for numerical computation, since the integrand decreases very quickly.

Integration over log-gamma

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The integral $\int _{0}^{z}\operatorname {log\Gamma } (x)\,dx$ can be expressed in terms of theBarnesG-function^[45]^[46] (seeBarnesG-function for a proof): $\int _{0}^{z}\operatorname {log\Gamma } (x)\,dx={\frac {z}{2}}\log(2\pi )+{\frac {z(1-z)}{2}}+z\operatorname {log\Gamma } (z)-\log G(z+1)$ whereRe(z) > −1.

It can also be written in terms of theHurwitz zeta function:^[47]^[48] $\int _{0}^{z}\operatorname {log\Gamma } (x)\,dx={\frac {z}{2}}\log(2\pi )+{\frac {z(1-z)}{2}}-\zeta '(-1)+\zeta '(-1,z).$

When $z=1$ it follows that $\int _{0}^{1}\operatorname {log\Gamma } (x)\,dx={\frac {1}{2}}\log(2\pi ),$ and this is a consequence ofRaabe's formula as well. O. Espinosa and V. Moll derived a similar formula for the integral of the square of $\operatorname {log\Gamma }$ :^[49] $\int _{0}^{1}\log ^{2}\Gamma (x)dx={\frac {\gamma ^{2}}{12}}+{\frac {\pi ^{2}}{48}}+{\frac {1}{3}}\gamma L_{1}+{\frac {4}{3}}L_{1}^{2}-\left(\gamma +2L_{1}\right){\frac {\zeta ^{\prime }(2)}{\pi ^{2}}}+{\frac {\zeta ^{\prime \prime }(2)}{2\pi ^{2}}},$ where $L_{1}$ is ${\frac {1}{2}}\log(2\pi )$ .

D. H. Bailey and his co-authors^[50] gave an evaluation for $L_{n}:=\int _{0}^{1}\log ^{n}\Gamma (x)\,dx$ when $n=1,2$ in terms of the Tornheim–Witten zeta function and its derivatives.

In addition, it is also known that^[51] $\lim _{n\to \infty }{\frac {L_{n}}{n!}}=1.$

Approximations

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Comparison of the gamma function (blue line) with the factorial (blue dots) and Stirling's approximation (red line)

Complex values of the gamma function can be approximated usingStirling's approximation or theLanczos approximation, $\Gamma (z)\sim {\sqrt {2\pi }}z^{z-1/2}e^{-z}\quad {\hbox{as }}z\to \infty {\hbox{ in }}\left|\arg(z)\right|<\pi .$ This is precise in the sense that the ratio of the approximation to the true value approaches 1 in the limit as $|z|\rightarrow \infty$ .

The gamma function can be computed to fixed precision for $\Re (z)\in [1,2]$ by applyingintegration by parts to Euler's integral. For any positive number x the gamma function can be written ${\begin{aligned}\Gamma (z)&=\int _{0}^{x}e^{-t}t^{z}\,{\frac {dt}{t}}+\int _{x}^{\infty }e^{-t}t^{z}\,{\frac {dt}{t}}\\&=x^{z}e^{-x}\sum _{n=0}^{\infty }{\frac {x^{n}}{z(z+1)\cdots (z+n)}}+\int _{x}^{\infty }e^{-t}t^{z}\,{\frac {dt}{t}}.\end{aligned}}$

WhenRe(z) ∈ [1,2] and $x\geq 1$ , the absolute value of the last integral is smaller than $(x+1)e^{-x}$ . By choosing a large enough⁠ $x {\displaystyle x}$ ⁠, this last expression can be made smaller than $2^{-N}$ for any desired value⁠ $N {\displaystyle N}$ ⁠. Thus, the gamma function can be evaluated to $N {\displaystyle N}$ bits of precision with the above series.

A fast algorithm for calculation of the Euler gamma function for any algebraic argument (including rational) was constructed by E.A. Karatsuba.^[52]^[53]^[54]

For arguments that are integer multiples of⁠1/24⁠, the gamma function can also be evaluated quickly usingarithmetic–geometric mean iterations (seeparticular values of the gamma function).^[55]

Practical implementations

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Unlike many other functions, such as aNormal Distribution, no obvious fast, accurate implementation that is easy to implement for the Gamma Function $\Gamma (z)$ is easily found. Therefore, it is worth investigating potential solutions. For the case that speed is more important than accuracy, published tables for $\Gamma (z)$ are easily found in an Internet search, such as the Online Wiley Library. Such tables may be used withlinear interpolation. Greater accuracy is obtainable with the use ofcubic interpolation at the cost of more computational overhead. Since $\Gamma (z)$ tables are usually published for argument values between 1 and 2, the property $\Gamma (z+1)=z\ \Gamma (z)$ may be used to quickly and easily translate all real values $z<1$ and $z>2$ into the range $1\leq z\leq 2$ , such that only tabulated values of $z {\displaystyle z}$ between 1 and 2 need be used.^[56]

If interpolation tables are not desirable, then theLanczos approximation mentioned above works well for 1 to 2 digits of accuracy for small, commonly used values of z. If the Lanczos approximation is not sufficiently accurate, theStirling's formula for the Gamma Function may be used.

Applications

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One author describes the gamma function as "Arguably, the most commonspecial function, or the least 'special' of them. The other transcendental functions [...] are called 'special' because you could conceivably avoid some of them by staying away from many specialized mathematical topics. On the other hand, the gamma function $\Gamma (z)$ is most difficult to avoid."^[57]

Integration problems

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The gamma function finds application in such diverse areas asquantum physics,astrophysics andfluid dynamics.^[58] Thegamma distribution, which is formulated in terms of the gamma function, is used instatistics to model a wide range of processes; for example, the time between occurrences of earthquakes.^[59]

The primary reason for the gamma function's usefulness in such contexts is the prevalence of expressions of the type $f(t)e^{-g(t)}$ which describe processes that decay exponentially in time or space. Integrals of such expressions can occasionally be solved in terms of the gamma function when no elementary solution exists. For example, if $f {\displaystyle f}$ is a power function and $g {\displaystyle g}$ is a linear function, a simple change of variables $u:=a\cdot t$ gives the evaluation

$\int _{0}^{\infty }t^{b}\,e^{-at}\,dt={\frac {1}{a^{b}}}\int _{0}^{\infty }u^{b}\,e^{-u}\,d\left({\frac {u}{a}}\right)={\frac {\Gamma (b+1)}{a^{b+1}}}.$

The fact that the integration is performed along the entire positive real line might signify that the gamma function describes the cumulation of a time-dependent process that continues indefinitely, or the value might be the total of a distribution in an infinite space.

It is of course frequently useful to take limits of integration other than $0 {\displaystyle 0}$ and $\infty$ to describe the cumulation of a finite process, in which case the ordinary gamma function is no longer a solution; the solution is then called anincomplete gamma function. (The ordinary gamma function, obtained by integrating across the entire positive real line, is sometimes called thecomplete gamma function for contrast.)

An important category of exponentially decaying functions is that ofGaussian functions $ae^{-{\frac {(x-b)^{2}}{c^{2}}}}$ and integrals thereof, such as theerror function. There are many interrelations between these functions and the gamma function; notably, the factor ${\sqrt {\pi }}$ obtained by evaluating ${\textstyle \Gamma \left({\frac {1}{2}}\right)}$ is the "same" as that found in the normalizing factor of the error function and thenormal distribution.

The integrals discussed so far involvetranscendental functions, but the gamma function also arises from integrals of purely algebraic functions. In particular, thearc lengths ofellipses and of thelemniscate, which are curves defined by algebraic equations, are given byelliptic integrals that in special cases can be evaluated in terms of the gamma function. The gamma function can also be used tocalculate "volume" and "area" of $n {\displaystyle n}$ -dimensionalhyperspheres.

Calculating products

[edit]

The gamma function's ability to generalize factorial products immediately leads to applications in many areas of mathematics; incombinatorics, and by extension in areas such asprobability theory and the calculation ofpower series. Many expressions involving products of successive integers can be written as some combination of factorials, the most important example perhaps being that of thebinomial coefficient. For example, for any complex numbers $z {\displaystyle z}$ and $n {\displaystyle n}$ , with $|z|<1$ , we can write $(1+z)^{n}=\sum _{k=0}^{\infty }{\frac {\Gamma (n+1)}{k!\Gamma (n-k+1)}}z^{k},$ which closely resembles the binomial coefficient when $n {\displaystyle n}$ is a non-negative integer, $(1+z)^{n}=\sum _{k=0}^{n}{\frac {n!}{k!(n-k)!}}z^{k}=\sum _{k=0}^{n}{\binom {n}{k}}z^{k}.$

The example of binomial coefficients motivates why the properties of the gamma function when extended to negative numbers are natural. A binomial coefficient gives the number of ways to choose $k {\displaystyle k}$ elements from a set of $n {\displaystyle n}$ elements; if $k>n$ , there are of course no ways. If $k>n$ , then $(n-k)!$ is the factorial of a negative integer and hence infinite if we use the gamma function definition of factorials—dividing by infinity gives the expected value of 0.

We can replace the factorial by a gamma function to extend any such formula to the complex numbers. Generally, this works for any product wherein each factor is arational function of the index variable, by factoring the rational function into linear expressions. If $P {\displaystyle P}$ and $Q {\displaystyle Q}$ are monic polynomials of degree $m {\displaystyle m}$ and $n {\displaystyle n}$ with respective roots $p_{1},\cdots ,p_{m}$ and $q_{1},\cdots ,q_{m}$ , we have $\prod _{i=a}^{b}{\frac {P(i)}{Q(i)}}=\left(\prod _{j=1}^{m}{\frac {\Gamma (b-p_{j}+1)}{\Gamma (a-p_{j})}}\right)\left(\prod _{k=1}^{n}{\frac {\Gamma (a-q_{k})}{\Gamma (b-q_{k}+1)}}\right).$

If we have a way to calculate the gamma function numerically, it is very simple to calculate numerical values of such products. The number of gamma functions in the right-hand side depends only on the degree of the polynomials, so it does not matter whether $b-a$ equals 5 or 10⁵. By taking the appropriate limits, the equation can also be made to hold even when the left-hand product contains zeros or poles.

By taking limits, certain rational products with infinitely many factors can be evaluated in terms of the gamma function as well. Due to theWeierstrass factorization theorem, analytic functions can be written as infinite products, and these can sometimes be represented as finite products or quotients of the gamma function. We have already seen one striking example: the reflection formula essentially represents the sine function as the product of two gamma functions. Starting from this formula, the exponential function as well as all the trigonometric and hyperbolic functions can be expressed in terms of the gamma function.

More functions yet, including thehypergeometric function and special cases thereof, can be represented by means of complexcontour integrals of products and quotients of the gamma function, calledMellin–Barnes integrals.

Analytic number theory

[edit]

An application of the gamma function is the study of theRiemann zeta function. A fundamental property of the Riemann zeta function is itsfunctional equation: $\Gamma \left({\frac {s}{2}}\right)\,\zeta (s)\,\pi ^{-{\frac {s}{2}}}=\Gamma \left({\frac {1-s}{2}}\right)\,\zeta (1-s)\,\pi ^{-{\frac {1-s}{2}}}.$

Among other things, this provides an explicit form for theanalytic continuation of the zeta function to a meromorphic function in the complex plane and leads to an immediate proof that the zeta function has infinitely many so-called "trivial" zeros on the real line.^[60] Another powerful formula is $\zeta (s)\;\Gamma (s)=\int _{0}^{\infty }{\frac {t^{s}}{e^{t}-1}}\,{\frac {dt}{t}}.$

Both formulas were derived byBernhard Riemann in his seminal 1859 paper "Ueber die Anzahl der Primzahlen unter einer gegebenen Größe" ("On the Number of Primes Less Than a Given Magnitude"), one of the milestones in the development ofanalytic number theory—the branch of mathematics that studiesprime numbers using the tools of mathematical analysis.

History

[edit]

The gamma function has caught the interest of some of the most prominent mathematicians of all time. Its history, notably documented byPhilip J. Davis in an article that won him the 1963Chauvenet Prize, reflects many of the major developments within mathematics since the 18th century. In the words of Davis, "each generation has found something of interest to say about the gamma function. Perhaps the next generation will also."^[1]

18th century: Euler and Stirling

[edit]

Daniel Bernoulli's letter toChristian Goldbach (Oct 6, 1729)

The problem of extending the factorial to non-integer arguments was apparently first considered byDaniel Bernoulli andChristian Goldbach in the 1720s. In particular, in a letter from Bernoulli to Goldbach dated 6 October 1729 Bernoulli introduced the product representation^[61] $x!=\lim _{n\to \infty }\left(n+1+{\frac {x}{2}}\right)^{x-1}\prod _{k=1}^{n}{\frac {k+1}{k+x}},$ which is well defined for real values ofx other than the negative integers.

Leonhard Euler later gave two different definitions: the first was not his integral but aninfinite product that is well defined for all complex numbersn other than the negative integers, $n!=\prod _{k=1}^{\infty }{\frac {\left(1+{\frac {1}{k}}\right)^{n}}{1+{\frac {n}{k}}}}\,,$ of which he informed Goldbach in a letter dated 13 October 1729. He wrote to Goldbach again on 8 January 1730, to announce his discovery of the integral representation $n!=\int _{0}^{1}(-\log s)^{n}\,ds,$ which is valid when the real part of the complex numbern is strictly greater than $-1$ (i.e., $\Re (n)>-1$ ). By the change of variables $t=-\ln s$ , this becomes the familiar Euler integral. Euler published his results in the paper "De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt" ("On transcendental progressions, that is, those whose general terms cannot be given algebraically"), submitted to theSt. Petersburg Academy on 28 November 1729.^[62] Euler further discovered some of the gamma function's important functional properties, including the reflection formula.

James Stirling, a contemporary of Euler, also attempted to find a continuous expression for the factorial and came up with what is now known asStirling's formula. Although Stirling's formula gives a good estimate of $n! {\displaystyle n!}$ , also for non-integers, it does not provide the exact value. Extensions of his formula that correct the error were given by Stirling himself and byJacques Philippe Marie Binet.

19th century: Gauss, Weierstrass, and Legendre

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De progressionibus transcendentibus, seu quarum termini generales algebraicae dari nequeunt — The first page of Euler's paper

Carl Friedrich Gauss rewrote Euler's product as $\Gamma (z)=\lim _{m\to \infty }{\frac {m^{z}m!}{z(z+1)(z+2)\cdots (z+m)}}$ and used this formula to discover new properties of the gamma function. Although Euler was a pioneer in the theory of complex variables, he does not appear to have considered the factorial of a complex number, as instead Gauss first did.^[63] Gauss also proved themultiplication theorem of the gamma function and investigated the connection between the gamma function andelliptic integrals.

Karl Weierstrass further established the role of the gamma function incomplex analysis, starting from yet another product representation, $\Gamma (z)={\frac {e^{-\gamma z}}{z}}\prod _{k=1}^{\infty }\left(1+{\frac {z}{k}}\right)^{-1}e^{\frac {z}{k}},$ where $\gamma$ is theEuler–Mascheroni constant. Weierstrass originally wrote his product as one for ${\frac {1}{\Gamma }}$ , in which case it is taken over the function's zeros rather than its poles. Inspired by this result, he proved what is known as theWeierstrass factorization theorem—that any entire function can be written as a product over its zeros in the complex plane; a generalization of thefundamental theorem of algebra.

The name gamma function and the symbol $\Gamma$ were introduced byAdrien-Marie Legendre around 1811; Legendre also rewrote Euler's integral definition in its modern form. Although the symbol is an upper-case Greek gamma, there is no accepted standard for whether the function name should be written "gamma function" or "Gamma function" (some authors simply write " $\Gamma$ -function"). The alternative "pi function" notation $\Pi (z)=z!$ due to Gauss is sometimes encountered in older literature, but Legendre's notation is dominant in modern works.

It is justified to ask why we distinguish between the "ordinary factorial" and the gamma function by using distinct symbols, and particularly why the gamma function should be normalized to $\Gamma (n+1)=n!$ instead of simply using $\Gamma (n)=n!$ . Consider that the notation for exponents, $x^{n}$ , has been generalized from integers to complex numbers $x^{z}$ without any change. Legendre's motivation for the normalization is not known, and has been criticized as cumbersome by some (the 20th-century mathematicianCornelius Lanczos, for example, called it "void of any rationality" and would instead use $z! {\displaystyle z!}$ ).^[64] Legendre's normalization does simplify some formulae, but complicates others. From a modern point of view, the Legendre normalization of the gamma function is the integral of the additivecharacter $e^{-x}$ against the multiplicative character $x^{z}$ with respect to theHaar measure ${\textstyle {\dfrac {dx}{x}}}$ on theLie group $\mathbb {R} ^{+}$ . Thus this normalization makes it clearer that the gamma function is a continuous analogue of aGauss sum.^[65]

19th–20th centuries: characterizing the gamma function

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It is somewhat problematic that a large number of definitions have been given for the gamma function. Although they describe the same function, it is not entirely straightforward to prove the equivalence. Stirling never proved that his extended formula corresponds exactly to Euler's gamma function; a proof was first given byCharles Hermite in 1900.^[66] Instead of finding a specialized proof for each formula, it would be desirable to have a general method of identifying the gamma function.

One way to prove equivalence would be to find adifferential equation that characterizes the gamma function. Most special functions in applied mathematics arise as solutions to differential equations, whose solutions are unique. However, the gamma function does not appear to satisfy any simple differential equation.Otto Hölder proved in 1887 that the gamma function at least does not satisfy anyalgebraic differential equation by showing that a solution to such an equation could not satisfy the gamma function's recurrence formula, making it atranscendentally transcendental function. This result is known asHölder's theorem.

A definite and generally applicable characterization of the gamma function was not given until 1922.Harald Bohr andJohannes Mollerup then proved what is known as theBohr–Mollerup theorem: that the gamma function is the unique solution to the factorialrecurrence relation that is positive andlogarithmically convex for positivez and whose value at 1 is 1 (a function is logarithmically convex if its logarithm is convex). Another characterisation is given by theWielandt theorem.

The Bohr–Mollerup theorem is useful because it is relatively easy to prove logarithmic convexity for any of the different formulas used to define the gamma function. Taking things further, instead of defining the gamma function by any particular formula, we can choose the conditions of the Bohr–Mollerup theorem as the definition, and then pick any formula we like that satisfies the conditions as a starting point for studying the gamma function. This approach was used by theBourbaki group.

Borwein & Corless review three centuries of work on the gamma function.^[67]

Reference tables and software

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A hand-drawn graph of the absolute value of the complex gamma function, fromTables of Higher Functions byJahnke andEmde [de]

Although the gamma function can be calculated virtually as easily as any mathematically simpler function with a modern computer—even with a programmable pocket calculator—this was of course not always the case. Until the mid-20th century, mathematicians relied on hand-made tables; in the case of the gamma function, notably a table computed by Gauss in 1813 and one computed by Legendre in 1825.^[68]

Tables of complex values of the gamma function, as well as hand-drawn graphs, were given inTables of Functions With Formulas and Curves byJahnke andEmde [de], first published in Germany in 1909. According toMichael Berry, "the publication in J&E of a three-dimensional graph showing the poles of the gamma function in the complex plane acquired an almost iconic status."^[69]

There was in fact little practical need for anything but real values of the gamma function until the 1930s, when applications for the complex gamma function were discovered in theoretical physics. As electronic computers became available for the production of tables in the 1950s, several extensive tables for the complex gamma function were published to meet the demand, including a table accurate to 12 decimal places from the U.S.National Bureau of Standards.^[1]

Reproduction of a famous complex plot by Janhke and Emde (Tables of Functions with Formulas and Curves, 4th ed., Dover, 1945) of the gamma function from −4.5 − 2.5i to 4.5 + 2.5i — Reproduction of a famous complex plot by Janhke and Emde (*Tables of Functions with Formulas and Curves*, 4th ed., Dover, 1945) of the gamma function from $-4.5-2.5i$ to $4.5+2.5i$ .

{\displaystyle -4.5-2.5i} — Reproduction of a famous complex plot by Janhke and Emde (*Tables of Functions with Formulas and Curves*, 4th ed., Dover, 1945) of the gamma function from $-4.5-2.5i$ to $4.5+2.5i$ .

Double-precision floating-point implementations of the gamma function and its logarithm are now available in most scientific computing software and special functions libraries, for exampleTK Solver,Matlab,GNU Octave, and theGNU Scientific Library. The gamma function was also added to theC standard library (math.h). Arbitrary-precision implementations are available in mostcomputer algebra systems, such asMathematica andMaple.PARI/GP,MPFR and MPFUN contain free arbitrary-precision implementations. In somesoftware calculators, such theWindows Calculator andGNOME Calculator, the factorial function returns $\Gamma (x+1)$ when the input $x {\displaystyle x}$ is a non-integer value.^[70]^[71]