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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g0433102.png" />-function''
+
''$\Gamma$-function''
  
A transcendental function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g0433103.png" /> that extends the values of the factorial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g0433104.png" /> to any complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g0433105.png" />. It was introduced in 1729 by L. Euler in a letter to Ch. Goldbach, using the infinite product
+
{{MSC|33B15|33B20,33D05}}
 +
{{TEX|done}}
 +
$
 +
\newcommand{\abs}[1]{\left|#1\right|}
 +
\newcommand{\Re}{\mathop{\mathrm{Re}}}
 +
\newcommand{\Im}{\mathop{\mathrm{Im}}}
 +
$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g0433106.png" /></td> </tr></table>
+
A  transcendental function $\Gamma(z)$ that extends the values of the  factorial $z!$ to any complex number $z$. It was introduced in 1729 by  L.&nbsp;Euler in a letter to Ch.&nbsp;Goldbach, using the  infinite product
 +
$$
 +
\Gamma(z) =
 +
\lim_{n\rightarrow\infty}\frac{n!n^z}{z(z+1)\ldots(z+n)} =
 +
\lim_{n\rightarrow\infty}\frac{n^z}{z(1+z/2)\ldots(1+z/n)},
 +
$$
 +
which  was used by L.&nbsp;Euler to obtain the integral representation  (Euler integral of the second kind, cf. [[Euler integrals]])
 +
$$
 +
\Gamma(z) = \int_0^\infty x^{z-1}e^{-x} \rd x,
 +
$$
 +
which  is valid for $\Re z > 0$. The multi-valuedness of the function  $x^{z-1}$ is eliminated by the formula $x^{z-1}=e^{(z-1)\ln x}$ with a  real $\ln x$. The symbol $\Gamma(z)$ and the name gamma-function were  proposed in 1814 by A.M.&nbsp;Legendre.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g0433107.png" /></td> </tr></table>
+
If $\Re z  < 0$ and $-k-1 < \Re z < -k$, $k=0,1,\ldots$, the gamma-function may be represented by the Cauchy–Saalschütz integral:
 
+
$$
which was used by L. Euler to obtain the integral representation (Euler integral of the second kind, cf. [[Euler integrals|Euler integrals]])
+
\Gamma(z) = \int_0^\infty x^{z-1}
 
+
\left(
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g0433108.png" /></td> </tr></table>
+
e^{-x} - \sum_{m=0}^k (-1)^m \frac{x^m}{m!}
 
+
\right) \rd x.
which is valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g0433109.png" />. The multi-valuedness of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331010.png" /> is eliminated by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331011.png" /> with a real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331012.png" />. The symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331013.png" /> and the name gamma-function were proposed in 1814 by A.M. Legendre.
+
$$
 
+
In the entire plane punctured at the points $z=0,-1,\ldots $, the gamma-function satisfies a Hankel integral representation:
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331016.png" /> the gamma-function may be represented by the Cauchy–Saalschütz integral:
+
$$
 
+
\Gamma(z) = \frac{1}{e^{2\pi iz} - 1} \int_C s^{z-1}e^{-s} \rd s,
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331017.png" /></td> </tr></table>
+
$$
 
+
where $s^{z-1} = e^{(z-1)\ln s}$ and $\ln s$ is the branch of the logarithm for which $0 < \arg\ln s < 2\pi$; the contour $C$ is represented in Fig. a. [FIXME] It is seen from the Hankel representation that $\Gamma(z)$ is a [[Meromorphic function|meromorphic function]]. At the points $z_n = -n$, $n=0,1,\ldots$ it has simple poles with residues $(-1)^n/n!$.
In the entire plane punctured at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331018.png" /> the gamma-function satisfies a Hankel integral representation:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331019.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331021.png" /> is the branch of the logarithm for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331022.png" />; the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331023.png" /> is represented in Fig. a. It is seen from the Hankel representation that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331024.png" /> is a [[Meromorphic function|meromorphic function]]. At the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331026.png" /> it has simple poles with residues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331027.png" />.
 
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310a.gif" />
Line 28: Line 39:
  
 
==Fundamental relations and properties of the gamma-function.==
 
==Fundamental relations and properties of the gamma-function.==
 
  
 
1) Euler's functional equation:
 
1) Euler's functional equation:
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331028.png" /></td> </tr></table>
+
z\Gamma(z) = \Gamma(z+1),
 
+
$$
 
or
 
or
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331029.png" /></td> </tr></table>
+
\Gamma(z) = \frac{1}{z\ldots(z+n)}\Gamma(z+n+1);
 
+
$$
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331031.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331032.png" /> is an integer; it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331033.png" />.
+
$\Gamma(1)=1$, $\Gamma(n+1) = n!$ if $n$ is an integer; it is assumed that $0! = \Gamma(1) = 1$.
  
 
2) Euler's completion formula:
 
2) Euler's completion formula:
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331034.png" /></td> </tr></table>
+
\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin \pi z}.
 
+
$$
In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331035.png" />;
+
In particular, $\Gamma(1/2)=\sqrt{\pi}$;
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331036.png" /></td> </tr></table>
+
\Gamma\left(n+\frac{1}{2}\right) =
 
+
\frac{1.3\ldots(2n-1)}{2^n}\sqrt{\pi}
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331037.png" /> is an integer;
+
$$
 
+
if $n>0$ is an integer;  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331038.png" /></td> </tr></table>
+
$$
 +
\abs{\Gamma\left(\frac{1}{2} + iy\right)}^2 =
 +
\frac{\pi}{\cosh y\pi},
 +
$$
 +
where $y$ is real.
  
 
3) Gauss' multiplication formula:
 
3) Gauss' multiplication formula:
 +
$$
 +
\prod_{k=0}^{m-1} \Gamma\left( z + \frac{k}{m} \right) =
 +
(2\pi)^{(m-1)/2}m^{(1/2)-mz}\Gamma(mz), \quad m = 2,3,\ldots
 +
$$
 +
If $m=2$, this is the Legendre duplication formula.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331039.png" /></td> </tr></table>
+
4) If $\Re z \geq \delta > 0$ or $\abs{\Im z} \geq \delta > 0$, then $\ln\Gamma(z)$ can be asymptotically expanded into the Stirling series:
 
+
$$
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331040.png" />, this is the Legendre duplication formula.
+
\ln\Gamma(z)
 
+
= \left(z-\frac{1}{2}\right)\ln z
4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331041.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331042.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331043.png" /> can be asymptotically expanded into the Stirling series:
+
- z
 
+
+ \frac{1}{2}\ln 2\pi
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331044.png" /></td> </tr></table>
+
+ \sum_{n=1}^m \frac{B_{2n}}{2n(2n-1)z^{2n-1}}
 
+
+ O\bigl(z^{-2m-1}\bigr), \quad m = 1,2,\ldots,
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331045.png" /></td> </tr></table>
+
$$
 
+
where $B_{2n}$ are the [[Bernoulli numbers]]. It implies the equality
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331046.png" /> are the [[Bernoulli numbers|Bernoulli numbers]]. It implies the equality
+
$$
 
+
\Gamma(z) = \sqrt{2\pi} z^{z-1/2} z^{-z}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331047.png" /></td> </tr></table>
+
\left(
 
+
1
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331048.png" /></td> </tr></table>
+
+ \frac{1}{12}z^{-1}
 
+
+ \frac{1}{288}z^{-2}
 +
- \frac{139}{51840}z^{-3}
 +
- \frac{571}{2488320}z^{-4}
 +
+ O\bigl(z^{-5}\bigr)
 +
\right).
 +
$$
 
In particular,
 
In particular,
 +
$$
 +
\Gamma(1+x) = \sqrt{2\pi} x^{x+1/2} e^{-x + \theta/12x},
 +
\quad 0 < \theta < 1.
 +
$$
 +
More accurate is Sonin's formula {{Cite|So}}:
 +
$$
 +
\Gamma(1+x) = \sqrt{2\pi} x^{x+1/2} e^{-x + 1/12(x+\theta)},
 +
\quad 0 < \theta < 1/2.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331049.png" /></td> </tr></table>
+
5) In the real domain, $\Gamma(x) > 0$ for $x > 0$ and it assumes the sign $(-1)^{k+1}$ on the segments $-k-1 < x < -k$, $k = 0,1,\ldots$ (Fig. b).
 
 
More accurate is Sonin's formula [[#References|[6]]]:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331050.png" /></td> </tr></table>
 
 
 
5) In the real domain, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331051.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331052.png" /> and it assumes the sign <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331053.png" /> on the segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331055.png" /> (Fig. b).
 
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310b.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310b.gif" />
Line 84: Line 111:
 
Figure: g043310b
 
Figure: g043310b
  
The graph of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331056.png" />.
+
The graph of the function $ $.
 
 
For all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331057.png" /> the inequality
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331058.png" /></td> </tr></table>
+
For all real $x$ the inequality
 +
$$
 +
\Gamma\Gamma^{\prime\prime} > \bigl(\Gamma^\prime\bigr)^2 \geq 0
 +
$$
 +
is  valid, i.e. all branches of both $\abs{\Gamma(x)}$ and  $\ln\abs{\Gamma(x)}$ are convex functions. The property of logarithmic  convexity defines the gamma-function among all solutions of the  functional equation
 +
$$
 +
\Gamma(1+x) = x\Gamma(x)
 +
$$
 +
up to a constant factor (see also the
 +
[[Bohr-Mollerup theorem|Bohr&ndash;Mollerup theorem]]).
  
is valid, i.e. all branches of both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331060.png" /> are convex functions. The property of logarithmic convexity defines the gamma-function among all solutions of the functional equation
+
For positive values of $x$ the gamma-function has a unique minimum at $x=1.4616321\ldots$ equal to $0.885603\ldots$. The local minima of the function $\abs{\Gamma(x)}$ form a sequence tending to zero as $x\rightarrow -\infty$.
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331061.png" /></td> </tr></table>
 
 
 
up to a constant factor.
 
 
 
For positive values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331062.png" /> the gamma-function has a unique minimum at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331063.png" /> equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331064.png" />. The local minima of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331065.png" /> form a sequence tending to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331066.png" />.
 
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310c.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310c.gif" />
Line 102: Line 130:
 
Figure: g043310c
 
Figure: g043310c
  
The graph of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331067.png" />.
+
The graph of the function $ $.
 
 
6) In the complex domain, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331068.png" />, the gamma-function rapidly decreases as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331069.png" />,
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331070.png" /></td> </tr></table>
 
 
 
7) The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331071.png" /> (Fig. c) is an entire function of order one and of maximal type; asymptotically, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331072.png" />,
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331073.png" /></td> </tr></table>
+
6) In the complex domain, if $\Re z > 0$, the gamma-function rapidly decreases as $\abs{\Im z} \rightarrow \infty$,
 +
$$
 +
\lim_{\abs{\Im z} \rightarrow \infty}
 +
\abs{\Gamma(z)}\abs{\Im z}^{(1/2)-\Re z}e^{\pi\abs{\Im z}/2} =
 +
\sqrt{2\pi}.
 +
$$
  
 +
7)  The function $1/\Gamma(z)$ (Fig. c) is an entire function of order one  and of maximal type; asymptotically, as $r \rightarrow \infty$,
 +
$$
 +
\ln M(r) \sim r \ln r,
 +
$$
 
where
 
where
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331074.png" /></td> </tr></table>
+
M(r) = \max_{\abs{z} = r} \frac{1}{\abs{\Gamma(z)}}.
 
+
$$
 
It can be represented by the infinite Weierstrass product:
 
It can be represented by the infinite Weierstrass product:
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331075.png" /></td> </tr></table>
+
\frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^\infty
 
+
\left(\left( 1 + \frac{z}{n} \right) e^{-z/n} \right),
which converges absolutely and uniformly on any compact set in the complex plane (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331076.png" /> is the [[Euler constant|Euler constant]]). A Hankel integral representation is valid:
+
$$
 
+
which converges absolutely and uniformly on any compact set in the complex plane ($\gamma$ is the [[Euler constant]]). A Hankel integral representation is valid:
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331077.png" /></td> </tr></table>
+
$$
 
+
\frac{1}{\Gamma(z)} = \frac{1}{2\pi i} \int_{C'} e^s s^{-z} \rd s,
where the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331078.png" /> is shown in Fig. d.
+
$$
 +
where the contour $C'$ is shown in Fig. d.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310d.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310d.gif" />
Line 130: Line 162:
 
Figure: g043310d
 
Figure: g043310d
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331079.png" />
+
$ $
 
 
G.F. Voronoi [[#References|[7]]] obtained integral representations for powers of the gamma-function.
 
 
 
In applications, the so-called poly gamma-functions — <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331080.png" />-th derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331081.png" /> — are of importance. The function (Gauss' <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331083.png" />-function)
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331084.png" /></td> </tr></table>
+
G.F.&nbsp;Voronoi {{Cite|Vo}} obtained integral representations for powers of the gamma-function.
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331085.png" /></td> </tr></table>
 
 
 
is meromorphic, has simple poles at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331086.png" /> and satisfies the functional equation
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331087.png" /></td> </tr></table>
 
 
 
The representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331088.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331089.png" /> yields the formula
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331090.png" /></td> </tr></table>
 
  
 +
In  applications, the so-called poly gamma-functions &mdash; $k$th  derivatives of $\ln\Gamma(z)$ &mdash; are of importance. The  function (Gauss' $\psi$-function)
 +
$$
 +
\psi(z) =
 +
\frac{\mathrm{d}}{\mathrm{d}z}\ln\Gamma(z) =
 +
\frac{\Gamma'(z)}{\Gamma(z)} =
 +
-\gamma + \sum_{n=0}^\infty \frac{z-1}{(n+1)(z+n)} =
 +
-\gamma + \int_0^1 \frac{1 - (1-t)^{z-1}}{t} \rd t
 +
$$
 +
is meromorphic, has simple poles at the points $z=0,-1,\ldots$ and satisfies the functional equation
 +
$$
 +
\psi(z+1) - \psi(z) = \frac{1}{z}.
 +
$$
 +
The representation of $\psi(z)$ for $\abs{z}<1$ yields the formula
 +
$$
 +
\ln\Gamma(1+z) =
 +
-\gamma z + \sum_{k=2}^\infty \frac{(-1)^k S_k}{k} z^k,
 +
$$
 
where
 
where
 +
$$
 +
S_k = \sum_{n=1}^\infty n^{-k}.
 +
$$
 +
This formula may be used to compute $\Gamma(z)$ in a neighbourhood of the point $z=1$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331091.png" /></td> </tr></table>
+
For other poly gamma-functions see {{Cite|BaEr}}. The [[Incomplete gamma-function|incomplete gamma-function]] is defined by the equation
 
+
$$
This formula may be used to compute <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331092.png" /> in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331093.png" />.
+
I(x,y) = \int_0^y e^{-t}t^{x-1} \rd t.
 
+
$$
For other poly gamma-functions see [[#References|[2]]]. The [[Incomplete gamma-function|incomplete gamma-function]] is defined by the equation
+
The functions $\Gamma(z)$ and $\psi(z)$ are transcendental functions which do not satisfy any linear differential equation with rational coefficients (Hölder's theorem).
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331094.png" /></td> </tr></table>
 
 
 
The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331095.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331096.png" /> are transcendental functions which do not satisfy any linear differential equation with rational coefficients (Hölder's theorem).
 
 
 
The exceptional importance of the gamma-function in mathematical analysis is due to the fact that it can be used to express a large number of definite integrals, infinite products and sums of series (see, for example, [[Beta-function|Beta-function]]). In addition, it is widely used in the theory of special functions (the [[Hypergeometric function|hypergeometric function]], of which the gamma-function is a limit case, [[Cylinder functions|cylinder functions]], etc.), in analytic number theory, etc.
 
  
====References====
+
The exceptional importance of the gamma-function in mathematical analysis is due to the fact that it can be used to express a large number of definite integrals, infinite products and sums of series (see, for example,  [[Beta-function]]). In addition, it is widely used in the theory of  special functions (the [[Hypergeometric function|hypergeometric function]], of which the gamma-function is a limit case, [[Cylinder functions|cylinder functions]], etc.), in analytic number theory, etc.
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press (1952)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''1. The gamma function. The hypergeometric functions. Legendre functions''' , McGraw-Hill (1953)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Functions of a real variable" , Addison-Wesley (1976) (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> , ''Math. anal., functions, limits, series, continued fractions'' , ''Handbook Math. Libraries'' , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N. Nielsen,  "Handbuch der Theorie der Gammafunktion" , Chelsea, reprint  (1965)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> N.Ya. Sonin,  "Studies on cylinder functions and special polynomials" , Moscow  (1954)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G.F. Voronoi,  "Studies of primitive parallelotopes" , ''Collected works'' , '''2''' , Kiev  (1952)  pp. 239–368  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  E. Jahnke,   F. Emde,  "Tables of functions with formulae and curves" , Dover, reprint  (1945) (Translated from German)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A. Angot,   "Compléments de mathématiques. A l'usage des ingénieurs de l'electrotechnique et des télécommunications" , C.N.E.T.  (1957)</TD></TR></table>
 
  
 +
====References====
  
 +
{|
 +
|-
 +
|valign="top"|{{Ref|An}}||valign="top"|  A. Angot, "Compléments de mathématiques. A l'usage des ingénieurs de  l'electrotechnique et des télécommunications", C.N.E.T. (1957)
 +
|-
 +
|valign="top"|{{Ref|BaEr}}||valign="top"|  H. Bateman (ed.)  A. Erdélyi (ed.), ''Higher transcendental  functions'', '''1. The gamma function. The hypergeometric functions.  Legendre functions''', McGraw-Hill (1953)
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"|  N. Bourbaki, "Elements of mathematics. Functions of a real variable",  Addison-Wesley (1976) (Translated from French)
 +
|-
 +
|valign="top"|{{Ref|JaEm}}||valign="top"|  E. Jahnke, F.  Emde, "Tables of functions with formulae and curves",  Dover, reprint (1945) (Translated from German)
 +
|-
 +
|valign="top"|{{Ref|Ni}}||valign="top"| N. Nielsen, "Handbuch der Theorie der Gammafunktion", Chelsea, reprint (1965)
 +
|-
 +
|valign="top"|{{Ref|So}}||valign="top"|  N.Ya. Sonin, "Studies on cylinder functions and special polynomials",  Moscow (1954) (In Russian)
 +
|-
 +
|valign="top"|{{Ref|Vo}}||valign="top"|  G.F. Voronoi, "Studies of primitive parallelotopes", ''Collected  works'', '''2''', Kiev (1952) pp. 239–368 (In Russian)
 +
|-
 +
|valign="top"|{{Ref|WhWa}}||valign="top"|  E.T. Whittaker, G.N. Watson, "A course of modern analysis", Cambridge  Univ. Press (1952)
 +
|-
 +
|}
  
====Comments====
+
====Comments====  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331098.png" />-analogue of the gamma-function is given by
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331099.png" /></td> </tr></table>
+
For  an approach to the gamma-function based on the [[Bohr-Mollerup  theorem|Bohr&ndash;Mollerup]] characterization, see the short  monograph by E.&nbsp;Artin {{Cite|Ar}}.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g043310100.png" /></td> </tr></table>
+
The $q$-analogue of the gamma-function is given by
 +
$$
 +
\Gamma_q(z) = (1-q)^{1-z}
 +
\prod_{k=1}^\infty \frac{1-q^{k+1}}{1-q^{k+z}}, \quad
 +
z \neq 0,-1,-2,\ldots;\quad 0<q<1,
 +
$$
 +
cf. {{Cite|As}}. Its origin goes back to E.&nbsp;Heine (1847) and D.&nbsp;Jackson (1904).
  
cf. [[#References|[a2]]]. Its origin goes back to E. Heine (1847) and D. Jackson (1904). For the gamma-function see also [[#References|[a1]]].
+
====References====
  
====References====
+
{|
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"E. Artin,   "The gamma function" , Holt, Rinehart &amp; Winston (1964)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Askey,   "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g043310101.png" />-Gamma and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g043310102.png" />-Beta functions" ''Appl. Anal.'' , '''8''' (1978) pp. 125–141</TD></TR></table>
+
|-
 +
|valign="top"|{{Ref|Ar}}||valign="top"| E. Artin, "The gamma function", Holt, Rinehart &amp; Winston (1964)
 +
|-
 +
|valign="top"|{{Ref|As}}||valign="top"| R. Askey, "The $q$-Gamma and $q$-Beta functions" ''Appl. Anal.'', '''8''' (1978) pp. 125–141
 +
|-
 +
|}

Revision as of 19:51, 27 April 2012

$\Gamma$-function

2010 Mathematics Subject Classification: Primary: 33B15 Secondary: 33B2033D05 [MSN][ZBL] $ \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\Re}{\mathop{\mathrm{Re}}} \newcommand{\Im}{\mathop{\mathrm{Im}}} $

A transcendental function $\Gamma(z)$ that extends the values of the factorial $z!$ to any complex number $z$. It was introduced in 1729 by L. Euler in a letter to Ch. Goldbach, using the infinite product $$ \Gamma(z) = \lim_{n\rightarrow\infty}\frac{n!n^z}{z(z+1)\ldots(z+n)} = \lim_{n\rightarrow\infty}\frac{n^z}{z(1+z/2)\ldots(1+z/n)}, $$ which was used by L. Euler to obtain the integral representation (Euler integral of the second kind, cf. Euler integrals) $$ \Gamma(z) = \int_0^\infty x^{z-1}e^{-x} \rd x, $$ which is valid for $\Re z > 0$. The multi-valuedness of the function $x^{z-1}$ is eliminated by the formula $x^{z-1}=e^{(z-1)\ln x}$ with a real $\ln x$. The symbol $\Gamma(z)$ and the name gamma-function were proposed in 1814 by A.M. Legendre.

If $\Re z < 0$ and $-k-1 < \Re z < -k$, $k=0,1,\ldots$, the gamma-function may be represented by the Cauchy–Saalschütz integral: $$ \Gamma(z) = \int_0^\infty x^{z-1} \left( e^{-x} - \sum_{m=0}^k (-1)^m \frac{x^m}{m!} \right) \rd x. $$ In the entire plane punctured at the points $z=0,-1,\ldots $, the gamma-function satisfies a Hankel integral representation: $$ \Gamma(z) = \frac{1}{e^{2\pi iz} - 1} \int_C s^{z-1}e^{-s} \rd s, $$ where $s^{z-1} = e^{(z-1)\ln s}$ and $\ln s$ is the branch of the logarithm for which $0 < \arg\ln s < 2\pi$; the contour $C$ is represented in Fig. a. [FIXME] It is seen from the Hankel representation that $\Gamma(z)$ is a meromorphic function. At the points $z_n = -n$, $n=0,1,\ldots$ it has simple poles with residues $(-1)^n/n!$.

Figure: g043310a

Fundamental relations and properties of the gamma-function.

1) Euler's functional equation: $$ z\Gamma(z) = \Gamma(z+1), $$ or $$ \Gamma(z) = \frac{1}{z\ldots(z+n)}\Gamma(z+n+1); $$ $\Gamma(1)=1$, $\Gamma(n+1) = n!$ if $n$ is an integer; it is assumed that $0! = \Gamma(1) = 1$.

2) Euler's completion formula: $$ \Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin \pi z}. $$ In particular, $\Gamma(1/2)=\sqrt{\pi}$; $$ \Gamma\left(n+\frac{1}{2}\right) = \frac{1.3\ldots(2n-1)}{2^n}\sqrt{\pi} $$ if $n>0$ is an integer; $$ \abs{\Gamma\left(\frac{1}{2} + iy\right)}^2 = \frac{\pi}{\cosh y\pi}, $$ where $y$ is real.

3) Gauss' multiplication formula: $$ \prod_{k=0}^{m-1} \Gamma\left( z + \frac{k}{m} \right) = (2\pi)^{(m-1)/2}m^{(1/2)-mz}\Gamma(mz), \quad m = 2,3,\ldots $$ If $m=2$, this is the Legendre duplication formula.

4) If $\Re z \geq \delta > 0$ or $\abs{\Im z} \geq \delta > 0$, then $\ln\Gamma(z)$ can be asymptotically expanded into the Stirling series: $$ \ln\Gamma(z) = \left(z-\frac{1}{2}\right)\ln z - z + \frac{1}{2}\ln 2\pi + \sum_{n=1}^m \frac{B_{2n}}{2n(2n-1)z^{2n-1}} + O\bigl(z^{-2m-1}\bigr), \quad m = 1,2,\ldots, $$ where $B_{2n}$ are the Bernoulli numbers. It implies the equality $$ \Gamma(z) = \sqrt{2\pi} z^{z-1/2} z^{-z} \left( 1 + \frac{1}{12}z^{-1} + \frac{1}{288}z^{-2} - \frac{139}{51840}z^{-3} - \frac{571}{2488320}z^{-4} + O\bigl(z^{-5}\bigr) \right). $$ In particular, $$ \Gamma(1+x) = \sqrt{2\pi} x^{x+1/2} e^{-x + \theta/12x}, \quad 0 < \theta < 1. $$ More accurate is Sonin's formula [So]: $$ \Gamma(1+x) = \sqrt{2\pi} x^{x+1/2} e^{-x + 1/12(x+\theta)}, \quad 0 < \theta < 1/2. $$

5) In the real domain, $\Gamma(x) > 0$ for $x > 0$ and it assumes the sign $(-1)^{k+1}$ on the segments $-k-1 < x < -k$, $k = 0,1,\ldots$ (Fig. b).

Figure: g043310b

The graph of the function $ $.

For all real $x$ the inequality $$ \Gamma\Gamma^{\prime\prime} > \bigl(\Gamma^\prime\bigr)^2 \geq 0 $$ is valid, i.e. all branches of both $\abs{\Gamma(x)}$ and $\ln\abs{\Gamma(x)}$ are convex functions. The property of logarithmic convexity defines the gamma-function among all solutions of the functional equation $$ \Gamma(1+x) = x\Gamma(x) $$ up to a constant factor (see also the Bohr–Mollerup theorem).

For positive values of $x$ the gamma-function has a unique minimum at $x=1.4616321\ldots$ equal to $0.885603\ldots$. The local minima of the function $\abs{\Gamma(x)}$ form a sequence tending to zero as $x\rightarrow -\infty$.

Figure: g043310c

The graph of the function $ $.

6) In the complex domain, if $\Re z > 0$, the gamma-function rapidly decreases as $\abs{\Im z} \rightarrow \infty$, $$ \lim_{\abs{\Im z} \rightarrow \infty} \abs{\Gamma(z)}\abs{\Im z}^{(1/2)-\Re z}e^{\pi\abs{\Im z}/2} = \sqrt{2\pi}. $$

7) The function $1/\Gamma(z)$ (Fig. c) is an entire function of order one and of maximal type; asymptotically, as $r \rightarrow \infty$, $$ \ln M(r) \sim r \ln r, $$ where $$ M(r) = \max_{\abs{z} = r} \frac{1}{\abs{\Gamma(z)}}. $$ It can be represented by the infinite Weierstrass product: $$ \frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^\infty \left(\left( 1 + \frac{z}{n} \right) e^{-z/n} \right), $$ which converges absolutely and uniformly on any compact set in the complex plane ($\gamma$ is the Euler constant). A Hankel integral representation is valid: $$ \frac{1}{\Gamma(z)} = \frac{1}{2\pi i} \int_{C'} e^s s^{-z} \rd s, $$ where the contour $C'$ is shown in Fig. d.

Figure: g043310d

$ $

G.F. Voronoi [Vo] obtained integral representations for powers of the gamma-function.

In applications, the so-called poly gamma-functions — $k$th derivatives of $\ln\Gamma(z)$ — are of importance. The function (Gauss' $\psi$-function) $$ \psi(z) = \frac{\mathrm{d}}{\mathrm{d}z}\ln\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z)} = -\gamma + \sum_{n=0}^\infty \frac{z-1}{(n+1)(z+n)} = -\gamma + \int_0^1 \frac{1 - (1-t)^{z-1}}{t} \rd t $$ is meromorphic, has simple poles at the points $z=0,-1,\ldots$ and satisfies the functional equation $$ \psi(z+1) - \psi(z) = \frac{1}{z}. $$ The representation of $\psi(z)$ for $\abs{z}<1$ yields the formula $$ \ln\Gamma(1+z) = -\gamma z + \sum_{k=2}^\infty \frac{(-1)^k S_k}{k} z^k, $$ where $$ S_k = \sum_{n=1}^\infty n^{-k}. $$ This formula may be used to compute $\Gamma(z)$ in a neighbourhood of the point $z=1$.

For other poly gamma-functions see [BaEr]. The incomplete gamma-function is defined by the equation $$ I(x,y) = \int_0^y e^{-t}t^{x-1} \rd t. $$ The functions $\Gamma(z)$ and $\psi(z)$ are transcendental functions which do not satisfy any linear differential equation with rational coefficients (Hölder's theorem).

The exceptional importance of the gamma-function in mathematical analysis is due to the fact that it can be used to express a large number of definite integrals, infinite products and sums of series (see, for example, Beta-function). In addition, it is widely used in the theory of special functions (the hypergeometric function, of which the gamma-function is a limit case, cylinder functions, etc.), in analytic number theory, etc.

References

[An] A. Angot, "Compléments de mathématiques. A l'usage des ingénieurs de l'electrotechnique et des télécommunications", C.N.E.T. (1957)
[BaEr] H. Bateman (ed.) A. Erdélyi (ed.), Higher transcendental functions, 1. The gamma function. The hypergeometric functions. Legendre functions, McGraw-Hill (1953)
[Bo] N. Bourbaki, "Elements of mathematics. Functions of a real variable", Addison-Wesley (1976) (Translated from French)
[JaEm] E. Jahnke, F. Emde, "Tables of functions with formulae and curves", Dover, reprint (1945) (Translated from German)
[Ni] N. Nielsen, "Handbuch der Theorie der Gammafunktion", Chelsea, reprint (1965)
[So] N.Ya. Sonin, "Studies on cylinder functions and special polynomials", Moscow (1954) (In Russian)
[Vo] G.F. Voronoi, "Studies of primitive parallelotopes", Collected works, 2, Kiev (1952) pp. 239–368 (In Russian)
[WhWa] E.T. Whittaker, G.N. Watson, "A course of modern analysis", Cambridge Univ. Press (1952)

Comments

For an approach to the gamma-function based on the Bohr–Mollerup characterization, see the short monograph by E. Artin [Ar].

The $q$-analogue of the gamma-function is given by $$ \Gamma_q(z) = (1-q)^{1-z} \prod_{k=1}^\infty \frac{1-q^{k+1}}{1-q^{k+z}}, \quad z \neq 0,-1,-2,\ldots;\quad 0<q<1, $$ cf. [As]. Its origin goes back to E. Heine (1847) and D. Jackson (1904).

References

[Ar] E. Artin, "The gamma function", Holt, Rinehart & Winston (1964)
[As] R. Askey, "The $q$-Gamma and $q$-Beta functions" Appl. Anal., 8 (1978) pp. 125–141
How to Cite This Entry:
Gamma-function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Gamma-function&oldid=25607
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article