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''$\Gamma$-function''
 
  
{{MSC|33B15|33B20,33D05}}
 
{{TEX|done}}
 
$
 
\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|meromorphic function]]. At the points $z_n = -n$, $n=0,1,\ldots$ it has simple poles with residues $(-1)^n/n!$.
 
 
<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:
 
$$
 
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 {{Cite|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).
 
 
<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 $x$ 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]]).
 
 
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 $x\rightarrow -\infty$.
 
 
<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 $\Re z > 0$, 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.
 
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310d.gif" />
 
 
Figure: g043310d
 
 
$ $
 
 
G.F.&nbsp;Voronoi {{Cite|Vo}} obtained integral representations for powers of the gamma-function.
 
 
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
 
$$
 
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 {{Cite|BaEr}}. The [[Incomplete gamma-function|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|hypergeometric function]], of which the gamma-function is a limit case, [[Cylinder functions|cylinder functions]], etc.), in analytic number theory, etc.
 
 
====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====
 
 
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}}.
 
 
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).
 
 
====References====
 
 
{|
 
|-
 
|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 18:51, 27 April 2012

How to Cite This Entry:
Gamma-function. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Gamma-function&oldid=25609