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''$\Gamma$-function''
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Post $\TeX$ remarks.
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* Added links to [[Bohr-Mollerup theorem|Bohr–Mollerup]]
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* Expanded the reference to Artin's monograph
 +
* Renamed the second integration contour from $C^*$ to $C'$
 +
* Used $\gamma$ for the Euler constant rather than $C$ (which also clashed with the notation for the first integration contour mentioned)
 +
* Redrew all figures
 +
--[[User:Jjg|Jjg]] 19:58, 27 April 2012 (CEST)
  
$
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: I have also used $\gamma$ for [[Euler constant]], since this is the modern convention. [[User:TBloom|TBloom]] 22:08, 27 April 2012 (CEST)
\newcommand{\Re}{\mathop{\mathrm{Re}}}
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:: Good idea, I have never seen anything but $\gamma$ in this context. According to this [http://mathworld.wolfram.com/Euler-MascheroniConstant.html article] on MathWorld, $C$ was used by Euler  (1735), $\gamma$ by Mascheroni (1790). So a not-so-modern modern convention :-) --[[User:Jjg|Jjg]] 22:34, 27 April 2012 (CEST)
$
 
 
 
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 $ $ 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>
 
 
 
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" />
 
 
 
Figure: g043310a
 
 
 
==Fundamental relations and properties of the gamma-function.==
 
 
 
1) Euler's functional equation:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
 
 
 
or
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
 
 
 
$ $, $ $ if $ $ is an integer; it is assumed that $ $.
 
 
 
2) Euler's completion formula:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
 
 
 
In particular, $ $;
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
 
 
 
if $ $ is an integer;
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
 
 
 
3) Gauss' multiplication formula:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
 
 
 
If $ $, this is the Legendre duplication formula.
 
 
 
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;">$ $</td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
 
 
 
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;">$ $</td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
 
 
 
In particular,
 
 
 
<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]]]:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
 
 
 
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" />
 
 
 
Figure: g043310b
 
 
 
The graph of the function $ $.
 
 
 
For all real $ $ the inequality
 
 
 
<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 $ $ 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;">$ $</td> </tr></table>
 
 
 
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 $ $.
 
 
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310c.gif" />
 
 
 
Figure: g043310c
 
 
 
The graph of the function $ $.
 
 
 
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;">$ $</td> </tr></table>
 
 
 
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;">$ $</td> </tr></table>
 
 
 
where
 
 
 
<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:
 
 
 
<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 ($ $ 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;">$ $</td> </tr></table>
 
 
 
where the contour $ $ is shown in Fig. d.
 
 
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310d.gif" />
 
 
 
Figure: g043310d
 
 
 
$ $
 
 
 
G.F. Voronoi [[#References|[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)
 
 
 
<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;">$ $</td> </tr></table>
 
 
 
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;">$ $</td> </tr></table>
 
 
 
The representation of $ $ for $ $ yields the formula
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;">$ $</td> </tr></table>
 
 
 
where
 
 
 
<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>
 
 
 
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|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
 
 
 
<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;">$ $</td> </tr></table>
 
 
 
cf.  [[#References|[a2]]]. Its origin goes back to E. Heine (1847) and D.  Jackson (1904). For the gamma-function see also[[#References|[a1]]].
 
 
 
====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>
 

Latest revision as of 16:10, 29 April 2012

Post $\TeX$ remarks.

  • Added links to Bohr–Mollerup
  • Expanded the reference to Artin's monograph
  • Renamed the second integration contour from $C^*$ to $C'$
  • Used $\gamma$ for the Euler constant rather than $C$ (which also clashed with the notation for the first integration contour mentioned)
  • Redrew all figures

--Jjg 19:58, 27 April 2012 (CEST)

I have also used $\gamma$ for Euler constant, since this is the modern convention. TBloom 22:08, 27 April 2012 (CEST)
Good idea, I have never seen anything but $\gamma$ in this context. According to this article on MathWorld, $C$ was used by Euler (1735), $\gamma$ by Mascheroni (1790). So a not-so-modern modern convention :-) --Jjg 22:34, 27 April 2012 (CEST)
How to Cite This Entry:
Gamma-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gamma-function&oldid=25555