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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g0433102.png" />-function''
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''$ $-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
 
  
<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 $ $ that extends the values of the factorial $ $ to any complex number $ $. It was introduced in 1729 by L. Euler in a letter to Ch. Goldbach, using the infinite 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/g0433107.png"  /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
which was  used by L. Euler to obtain the integral representation (Euler integral  of the second kind, cf. [[Euler integrals|Euler integrals]])
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
<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>
+
which was used by L. Euler to obtain the integral representation (Euler integral of the second kind, cf[[Euler integrals|Euler    integrals]])
  
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.
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
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:
+
which is valid for $ $. The multi-valuedness of the function $ $ is eliminated by the formula $ $ with a real $ $. The symbol $ $ and the name gamma-function were proposed in 1814 by A.M. 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/g04331017.png"  /></td> </tr></table>
+
If $ $ and $ $, $ $ the gamma-function may be represented by the Cauchy–Saalschütz integral:
  
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;">$ $</td> </tr></table>
  
<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>
+
In the entire plane punctured at the points $ $ the gamma-function satisfies a Hankel integral representation:
  
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" />.
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
 +
 
 +
where $ $ and $ $ is the branch of the logarithm for which $ $; the contour $ $ is represented in Fig. a. It is seen from the Hankel representation that $ $ is a [[Meromorphic function|meromorphic     function]]. At the points $ $, $ $ it has simple poles with residues $ $.
  
 
<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 29:
  
 
==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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
 
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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
<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" />.
+
$ $, $ $ if $ $ is an integer; it is assumed that $ $.
  
 
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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331035.png" />;
+
In particular, $ $;
  
<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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331037.png" /> is an integer;
+
if $ $ 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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
 
3) Gauss' multiplication formula:
 
3) Gauss' multiplication 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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
If <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331040.png" />, this is the Legendre duplication formula.
+
If $ $, this is the Legendre duplication formula.
  
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:
+
4) If $ $ or $ $, then $ $ can be asymptotically expanded into the Stirling series:
  
<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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
<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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
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
+
where $ $ are the [[Bernoulli numbers|Bernoulli numbers]]. It implies the equality
  
<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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
<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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
 
In particular,
 
In particular,
  
<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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
 
More accurate is Sonin's formula [[#References|[6]]]:
 
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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</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).
+
5) In the real domain, $ $ for $ $ and it assumes the sign $ $ on the segments $ $, $ $ (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 84:
 
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
+
For all real $ $ 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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
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
+
is valid, i.e. all branches of both $ $ and $ $ are convex functions. The property of logarithmic convexity defines the gamma-function among all solutions of 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/g04331061.png"  /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
 
up to a constant factor.
 
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" />.
+
For positive values of $ $ the gamma-function has a unique minimum at $ $ equal to $ $. The local minima of the function $ $ form a sequence tending to zero as $ $.
  
 
<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 102:
 
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" />,
+
6) In the complex domain, if $ $, the gamma-function rapidly decreases as $ $,
  
<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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</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" />,
+
7) The function $ $ (Fig. c) is an entire function of order one and of maximal type; asymptotically, as $ $,
  
<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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
 
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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
 
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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
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 ($ $ is the [[Euler constant|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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
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 $ $ 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 130:
 
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.
 
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)
+
In applications, the so-called poly gamma-functions — $ $-th derivatives of $ $ — are of importance. The function (Gauss' $ $-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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
<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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</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
+
is meromorphic, has simple poles at the points $ $ 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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</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
+
The representation of $ $ for $ $ 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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
 
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/g04331091.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
 +
 
 +
This formula may be used to compute $ $ in a neighbourhood of the point $ $.
 +
 
 +
For other poly gamma-functions see [[#References|[2]]]. The[[Incomplete gamma-function|incomplete gamma-function]] is defined by the equation
 +
 
 +
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
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" />.
+
The functions $ $ and $ $ are transcendental functions which do not satisfy any linear differential equation with rational coefficients (Hölder's theorem).
  
For  other poly gamma-functions see [[#References|[2]]]. The [[Incomplete  gamma-function|incomplete gamma-function]] is defined by the equation
+
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.
  
<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>
+
====References====  
  
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).
+
<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>
  
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====
 
<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>
 
  
 +
====Comments====
  
 +
The $ $-analogue of the gamma-function is given by
  
====Comments====
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
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>
+
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
  
<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>
+
cf. [[#References|[a2]]]. Its origin goes back to E. Heine (1847) and DJackson (1904). For the gamma-function see also[[#References|[a1]]].
  
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 $ $-Gamma and $ $-Beta functions" ''Appl. Anal.'' , '''8''' (1978) pp. 125–141</TD></TR></table>
<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>
 

Revision as of 23:37, 26 April 2012

$ $-function


A transcendental function $ $ that extends the values of the factorial $ $ to any complex number $ $. It was introduced in 1729 by L. Euler in a letter to Ch. Goldbach, using the infinite product

$ $
$ $

which was used by L. Euler to obtain the integral representation (Euler integral of the second kind, cf. Euler integrals)

$ $

which is valid for $ $. The multi-valuedness of the function $ $ is eliminated by the formula $ $ with a real $ $. The symbol $ $ and the name gamma-function were proposed in 1814 by A.M. Legendre.

If $ $ and $ $, $ $ the gamma-function may be represented by the Cauchy–Saalschütz integral:

$ $

In the entire plane punctured at the points $ $ the gamma-function satisfies a Hankel integral representation:

$ $

where $ $ and $ $ is the branch of the logarithm for which $ $; the contour $ $ is represented in Fig. a. It is seen from the Hankel representation that $ $ is a meromorphic function. At the points $ $, $ $ it has simple poles with residues $ $.

Figure: g043310a

Fundamental relations and properties of the gamma-function.

1) Euler's functional equation:

$ $

or

$ $

$ $, $ $ if $ $ is an integer; it is assumed that $ $.

2) Euler's completion formula:

$ $

In particular, $ $;

$ $

if $ $ is an integer;

$ $

3) Gauss' multiplication formula:

$ $

If $ $, this is the Legendre duplication formula.

4) If $ $ or $ $, then $ $ can be asymptotically expanded into the Stirling series:

$ $
$ $

where $ $ are the Bernoulli numbers. It implies the equality

$ $
$ $

In particular,

$ $

More accurate is Sonin's formula [6]:

$ $

5) In the real domain, $ $ for $ $ and it assumes the sign $ $ on the segments $ $, $ $ (Fig. b).

Figure: g043310b

The graph of the function $ $.

For all real $ $ the inequality

$ $

is valid, i.e. all branches of both $ $ and $ $ are convex functions. The property of logarithmic convexity defines the gamma-function among all solutions of the functional equation

$ $

up to a constant factor.

For positive values of $ $ the gamma-function has a unique minimum at $ $ equal to $ $. The local minima of the function $ $ form a sequence tending to zero as $ $.

Figure: g043310c

The graph of the function $ $.

6) In the complex domain, if $ $, the gamma-function rapidly decreases as $ $,

$ $

7) The function $ $ (Fig. c) is an entire function of order one and of maximal type; asymptotically, as $ $,

$ $

where

$ $

It can be represented by the infinite Weierstrass product:

$ $

which converges absolutely and uniformly on any compact set in the complex plane ($ $ is the Euler constant). A Hankel integral representation is valid:

$ $

where the contour $ $ is shown in Fig. d.

Figure: g043310d

$ $

G.F. Voronoi [7] obtained integral representations for powers of the gamma-function.

In applications, the so-called poly gamma-functions — $ $-th derivatives of $ $ — are of importance. The function (Gauss' $ $-function)

$ $
$ $

is meromorphic, has simple poles at the points $ $ and satisfies the functional equation

$ $

The representation of $ $ for $ $ yields the formula

$ $

where

$ $

This formula may be used to compute $ $ in a neighbourhood of the point $ $.

For other poly gamma-functions see [2]. Theincomplete gamma-function is defined by the equation

$ $

The functions $ $ and $ $ 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 (thehypergeometric function, of which the gamma-function is a limit case, cylinder functions, etc.), in analytic number theory, etc.

References

[1] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952)
[2] H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 1. The gamma function. The hypergeometric functions. Legendre functions , McGraw-Hill (1953)
[3] N. Bourbaki, "Elements of mathematics. Functions of a real variable" , Addison-Wesley (1976) (Translated from French)
[4] , Math. anal., functions, limits, series, continued fractions , Handbook Math. Libraries , Moscow (1961) (In Russian)
[5] N. Nielsen, "Handbuch der Theorie der Gammafunktion" , Chelsea, reprint (1965)
[6] N.Ya. Sonin, "Studies on cylinder functions and special polynomials" , Moscow (1954) (In Russian)
[7] G.F. Voronoi, "Studies of primitive parallelotopes" , Collected works , 2 , Kiev (1952) pp. 239–368 (In Russian)
[8] E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German)
[9] A. Angot, "Compléments de mathématiques. A l'usage des ingénieurs de l'electrotechnique et des télécommunications" , C.N.E.T. (1957)


Comments

The $ $-analogue of the gamma-function is given by

$ $
$ $

cf. [a2]. Its origin goes back to E. Heine (1847) and D. Jackson (1904). For the gamma-function see also[a1].

References

[a1] E. Artin, "The gamma function" , Holt, Rinehart & Winston (1964)
[a2] R. Askey, "The $ $-Gamma and $ $-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=25552