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''$\Gamma$-function''

{{MSC|33B15|33B20,33D05}}
{{TEX|done}}
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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.

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:
\begin{equation}
\label{eq1}
\Gamma(z) = \frac{1}{e^{2\pi iz} - 1} \int_C s^{z-1}e^{-s} \rd s,
\end{equation}
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 Figure&nbsp;1. 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  $(-1)^n/n!$.

<span id="Fig1">
[[File:Gamma-function-1.png| center|  frame|  Figure 1. The contour of integration $C$ of equation \ref{eq1}   ([[Media:Gamma-function-1.pdf|pdf]]) ]]
</span>

==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}}:
$$
\Gamma(1+x) = \sqrt{2\pi} x^{x+1/2} e^{-x + 1/12(x+\theta)}, 
\quad 0 < \theta < 1/2.
$$

<span id="Fig2">
[[File:Gamma-function-2.png| right|   frame|  Figure 2. The Gamma function on the real line    ([[Media:Gamma-function-2.pdf|pdf]]) ]]
</span>
5)  In the real 
domain, $\Gamma(x) > 0$ for $x > 0$ and it assumes  the sign $(-1)^{k+1}$ on the segments $-k-1 < x < -k$, $k =  0,1,\ldots$ (Figure&nbsp;2).

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
|-
|}