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| − | ''$\Gamma$-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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| − | | + | : 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) |
| − | \newcommand{\abs}[1]{\left|#1\right|}
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| − | \newcommand{\Re}{\mathop{\mathrm{Re}}}
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| − | \newcommand{\Im}{\mathop{\mathrm{Im}}}
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| − | $
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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. Euler in a letter to Ch. Goldbach, using the infinite product
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| − | $$
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| − | \Gamma(z) =
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| − | \lim_{n\rightarrow\infty}\frac{n!n^z}{z(z+1)\ldots(z+n)} =
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| − | \lim_{n\rightarrow\infty}\frac{n^z}{z(1+z/2)\ldots(1+z/n)},
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| − | $$
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| − | which was used by L. Euler to obtain the integral representation (Euler integral of the second kind, cf. [[Euler integrals]])
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| − | $$
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| − | \Gamma(z) = \int_0^\infty x^{z-1}e^{-x} \rd x,
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| − | $$
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| − | 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.
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| − | 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:
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| − | $$ | |
| − | \Gamma(z) = \int_0^\infty x^{z-1}
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| − | \left(
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| − | e^{-x} - \sum_{m=0}^k (-1)^m \frac{x^m}{m!}
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| − | \right) \rd x.
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| − | $$
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| − | In the entire plane punctured at the points $z=0,-1,\ldots $, the gamma-function satisfies a Hankel integral representation:
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| − | $$
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| − | \Gamma(z) = \frac{1}{e^{2\pi iz} - 1} \int_C s^{z-1}e^{-s} \rd s,
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| − | $$
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| − | 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 $(-1)^n/n!$.
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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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| − | $$
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| − | z\Gamma(z) = \Gamma(z+1),
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| − | $$
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| − | or
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| − | $$
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| − | \Gamma(z) = \frac{1}{z\ldots(z+n)}\Gamma(z+n+1);
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| − | $$
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| − | $\Gamma(1)=1$, $\Gamma(n+1) = n!$ if $n$ is an integer; it is assumed that $0! = \Gamma(1) = 1$.
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| − | | |
| − | 2) Euler's completion formula:
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| − | $$
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| − | \Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin \pi z}.
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| − | $$
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| − | In particular, $\Gamma(1/2)=\sqrt{\pi}$;
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| − | $$
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| − | \Gamma\left(n+\frac{1}{2}\right) =
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| − | \frac{1.3\ldots(2n-1)}{2^n}\sqrt{\pi}
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| − | $$
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| − | if $n>0$ is an integer;
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| − | $$
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| − | \abs{\Gamma\left(\frac{1}{2} + iy\right)}^2 =
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| − | \frac{\pi}{\cosh y\pi},
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| − | $$
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| − | where $y$ is real.
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| − | 3) Gauss' multiplication formula:
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| − | $$
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| − | \prod_{k=0}^{m-1} \Gamma\left( z + \frac{k}{m} \right) =
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| − | (2\pi)^{(m-1)/2}m^{(1/2)-mz}\Gamma(mz), \quad m = 2,3,\ldots
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| − | $$
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| − | If $m=2$, this is the Legendre duplication formula.
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| − | | |
| − | 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:
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| − | $$
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| − | \ln\Gamma(z)
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| − | = \left(z-\frac{1}{2}\right)\ln z
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| − | - z
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| − | + \frac{1}{2}\ln 2\pi
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| − | + \sum_{n=1}^m \frac{B_{2n}}{2n(2n-1)z^{2n-1}}
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| − | + O\bigl(z^{-2m-1}\bigr), \quad m = 1,2,\ldots,
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| − | $$
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| − | where $B_{2n}$ are the [[Bernoulli numbers]]. It implies the equality
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| − | $$
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| − | \Gamma(z) = \sqrt{2\pi} z^{z-1/2} z^{-z}
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| − | \left(
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| − | 1
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| − | + \frac{1}{12}z^{-1}
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| − | + \frac{1}{288}z^{-2}
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| − | - \frac{139}{51840}z^{-3}
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| − | - \frac{571}{2488320}z^{-4}
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| − | + O\bigl(z^{-5}\bigr)
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| − | \right).
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| − | $$
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| − | In particular,
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| − | $$
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| − | \Gamma(1+x) = \sqrt{2\pi} x^{x+1/2} e^{-x + \theta/12x},
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| − | \quad 0 < \theta < 1.
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| − | $$
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| − | More accurate is Sonin's formula [[#References|[6]]]:
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| − | $$
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| − | \Gamma(1+x) = \sqrt{2\pi} x^{x+1/2} e^{-x + 1/12(x+\theta)},
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| − | \quad 0 < \theta < 1/2.
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| − | $$
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| − | 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).
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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 $ $.
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| − | For all real $x$ the inequality
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| − | $$
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| − | \Gamma\Gamma^{\prime\prime} > \bigl(\Gamma^\prime\bigr)^2 \geq 0
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| − | $$
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| − | 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
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| − | $$
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| − | \Gamma(1+x) = x\Gamma(x)
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| − | $$
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| − | up to a constant factor (see also the
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| − | [[Bohr-Mollerup theorem|Bohr–Mollerup theorem]]).
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| − | 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 $x\rightarrow -\infty$.
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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 $ $.
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| − | 6) In the complex domain, if $\Re z > 0$, the gamma-function rapidly decreases as $\abs{\Im z} \rightarrow \infty$,
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| − | $$
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| − | \lim_{\abs{\Im z} \rightarrow \infty}
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| − | \abs{\Gamma(z)}\abs{\Im z}^{(1/2)-\Re z}e^{\pi\abs{\Im z}/2} =
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| − | \sqrt{2\pi}.
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| − | $$
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| − | 7) The function $1/\Gamma(z)$ (Fig. c) is an entire function of order one and of maximal type; asymptotically, as $r \rightarrow \infty$,
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| − | $$
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| − | \ln M(r) \sim r \ln r,
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| − | $$
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| − | where
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| − | $$
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| − | M(r) = \max_{\abs{z} = r} \frac{1}{\abs{\Gamma(z)}}.
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| − | $$
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| − | It can be represented by the infinite Weierstrass product:
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| − | $$
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| − | \frac{1}{\Gamma(z)} = z e^{\gamma z} \prod_{n=1}^\infty
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| − | \left(\left( 1 + \frac{z}{n} \right) e^{-z/n} \right),
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| − | $$
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| − | which converges absolutely and uniformly on any compact set in the complex plane ($\gamma$ is the [[Euler constant]]). A Hankel integral representation is valid:
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| − | $$
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| − | \frac{1}{\Gamma(z)} = \frac{1}{2\pi i} \int_{C'} e^s s^{-z} \rd s,
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| − | $$
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| − | where the contour $C'$ 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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| − | $ $
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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 — $k$th derivatives of $\ln\Gamma(z)$ — are of importance. The function (Gauss' $\psi$-function)
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| − | $$
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| − | \psi(z) =
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| − | \frac{\mathrm{d}}{\mathrm{d}z}\ln\Gamma(z) =
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| − | \frac{\Gamma'(z)}{\Gamma(z)} =
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| − | -\gamma + \sum_{n=0}^\infty \frac{z-1}{(n+1)(z+n)} =
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| − | -\gamma + \int_0^1 \frac{1 - (1-t)^{z-1}}{t} \rd t
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| − | $$
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| − | is meromorphic, has simple poles at the points $z=0,-1,\ldots$ and satisfies the functional equation
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| − | $$
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| − | \psi(z+1) - \psi(z) = \frac{1}{z}.
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| − | $$
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| − | The representation of $\psi(z)$ for $\abs{z}<1$ yields the formula
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| − | $$
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| − | \ln\Gamma(1+z) =
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| − | -\gamma z + \sum_{k=2}^\infty \frac{(-1)^k S_k}{k} z^k,
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| − | $$
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| − | where
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| − | $$
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| − | S_k = \sum_{n=1}^\infty n^{-k}.
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| − | $$
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| − | This formula may be used to compute $\Gamma(z)$ in a neighbourhood of the point $z=1$.
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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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| − | $$
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| − | I(x,y) = \int_0^y e^{-t}t^{x-1} \rd t.
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| − | $$
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| − | The functions $\Gamma(z)$ and $\psi(z)$ 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]]). 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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| − | For an approach to the gamma-function based on the [[Bohr-Mollerup theorem|Bohr–Mollerup]] characterization, see the short monograph by E. Artin [[#References|[a1]]].
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| − | The $q$-analogue of the gamma-function is given by
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| − | $$
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| − | \Gamma_q(z) = (1-q)^{1-z}
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| − | \prod_{k=1}^\infty \frac{1-q^{k+1}}{1-q^{k+z}}, \quad
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| − | z \neq 0,-1,-2,\ldots;\quad 0<q<1,
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| − | $$
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| − | cf. [[#References|[a2]]]. Its origin goes back to E. Heine (1847) and D. Jackson (1904).
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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 $ $-Gamma and $ $-Beta functions" ''Appl. Anal.'' , '''8''' (1978) pp. 125–141</TD></TR></table>
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