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''$\Gamma$-function''


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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:
$$
\Gamma(z) = \frac{1}{e^{2\pi iz} - 1} \int_C s^{z-1}e^{-s} \rd s,
$$
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 [[#References|[6]]]:
$$
\Gamma(1+x) = \sqrt{2\pi} x^{x+1/2} e^{-x + 1/12(x+\theta)}, 
\quad 0 < \theta < 1/2.
$$

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 $ $ (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>