Difference between revisions of "Barnes G"
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$$G(1+z)=(2\pi)^{\frac{z}{2}} \exp \left( - \dfrac{z+z^2(1+\gamma)}{2} \right) \displaystyle\prod_{k=1}^{\infty} \left\{ \left( 1+\dfrac{z}{k} \right)^k \exp \left( \dfrac{z^2}{2k}-z \right) \right\},$$ | $$G(1+z)=(2\pi)^{\frac{z}{2}} \exp \left( - \dfrac{z+z^2(1+\gamma)}{2} \right) \displaystyle\prod_{k=1}^{\infty} \left\{ \left( 1+\dfrac{z}{k} \right)^k \exp \left( \dfrac{z^2}{2k}-z \right) \right\},$$ | ||
where $\exp$ denotes the [[exponential function]] and $\gamma$ denotes the [[Euler-Mascheroni constant]]. | where $\exp$ denotes the [[exponential function]] and $\gamma$ denotes the [[Euler-Mascheroni constant]]. | ||
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| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Barnesgplot.png|Graph of $G$. | ||
| + | </gallery> | ||
| + | </div> | ||
=Properties= | =Properties= | ||
Revision as of 00:40, 24 May 2016
The Barnes $G$ function is defined by the following Weierstrass factorization: $$G(1+z)=(2\pi)^{\frac{z}{2}} \exp \left( - \dfrac{z+z^2(1+\gamma)}{2} \right) \displaystyle\prod_{k=1}^{\infty} \left\{ \left( 1+\dfrac{z}{k} \right)^k \exp \left( \dfrac{z^2}{2k}-z \right) \right\},$$ where $\exp$ denotes the exponential function and $\gamma$ denotes the Euler-Mascheroni constant.
Graph of $G$.
Properties
Theorem: The following formula holds: $$G(z+1)=\Gamma(z)G(z)$$ with normalization $G(1)=1$.
Proof: █
Corollary: The following values hold: $$G(n) = \left\{ \begin{array}{ll} 0 &; n=-1,-2,\ldots \\ \displaystyle\prod_{i=0}^{n-2} i!&; n=0,1,2,\ldots \end{array} \right.$$
Proof: █
