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| − | ''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g0433102.png" />-function''
| + | Post $\TeX$ remarks. |
| | + | * Added links to [[Bohr-Mollerup theorem|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 |
| | + | --[[User:Jjg|Jjg]] 19:58, 27 April 2012 (CEST) |
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| − | 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
| + | : I have also used $\gamma$ for [[Euler constant]], since this is the modern convention. [[User:TBloom|TBloom]] 22:08, 27 April 2012 (CEST) |
| − | | + | :: 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) |
| − | <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>
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| − | <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>
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| − | which was used by L. Euler to obtain the integral representation (Euler integral of the second kind, cf. [[Euler integrals|Euler integrals]])
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| − | <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>
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| − | 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.
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| − | 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:
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| − | <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>
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| − | 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:
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| − | <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>
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| − | 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" />.
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| − | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310a.gif" />
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| − | Figure: g043310a
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| − | ==Fundamental relations and properties of the gamma-function.==
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| − | 1) Euler's functional equation:
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| − | <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>
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| − | or
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| − | <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>
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| − | <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" />.
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| − | 2) Euler's completion formula:
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| − | <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>
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| − | In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331035.png" />;
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| − | <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>
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| − | if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331037.png" /> is an integer;
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| − | <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>
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| − | 3) Gauss' multiplication formula:
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| − | <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>
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| − | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331040.png" />, this is the Legendre duplication formula.
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| − | 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:
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| − | <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>
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| − | <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>
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| − | 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
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| − | <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>
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| − | <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>
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| − | In particular,
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| − | <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>
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| − | More accurate is Sonin's formula [[#References|[6]]]:
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| − | <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>
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| − | 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).
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| − | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310b.gif" />
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| − | Figure: g043310b
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| − | The graph of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331056.png" />.
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| − | For all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331057.png" /> the inequality
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| − | <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>
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| − | 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
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| − | <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>
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| − | up to a constant factor.
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| − | 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" />.
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| − | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310c.gif" />
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| − | Figure: g043310c
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| − | The graph of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331067.png" />.
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| − | 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" />,
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| − | <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>
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| − | 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" />,
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| − | <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>
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| − | where
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| − | <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>
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| − | It can be represented by the infinite Weierstrass product:
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| − | <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>
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| − | 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:
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| − | <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>
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| − | where the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331078.png" /> is shown in Fig. d.
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| − | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g043310d.gif" />
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| − | Figure: g043310d
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| − | <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043310/g04331079.png" />
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| − | G.F. Voronoi [[#References|[7]]] obtained integral representations for powers of the gamma-function.
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| − | 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)
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| − | <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>
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| − | <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>
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| − | 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
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| − | <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>
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| − | 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
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| − | <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>
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| − | where
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| − | <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>
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| − | 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" />.
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| − | For other poly gamma-functions see [[#References|[2]]]. The [[Incomplete gamma-function|incomplete gamma-function]] is defined by the equation
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| − | <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>
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| − | 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).
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| − | 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.
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| − | ====References====
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| − | <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>
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| − | ====Comments====
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| − | 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
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| − | <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>
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| − | <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>
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| − | cf. [[#References|[a2]]]. Its origin goes back to E. Heine (1847) and D. Jackson (1904). For the gamma-function see also [[#References|[a1]]].
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| − | ====References====
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Artin, "The gamma function" , Holt, Rinehart & 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>
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