Difference between revisions of "Barnes G"
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(Created page with "The Barnes $G$ function is defined by the following Weierstrass factorization: $$G(1+z)=(2\pi)^{\frac{z}{2}} \exp \left( - \dfrac{z+z^2(1+\gamma)}{2} \right) \displaystyle...") |
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$$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\},$$ | ||
where $\exp$ denotes the [[exponential function]] and $\gamma$ denotes the [[Euler-Mascheroni constant]]. | where $\exp$ denotes the [[exponential function]] and $\gamma$ denotes the [[Euler-Mascheroni constant]]. | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$G(z+1)=\Gamma(z)G(z)$$ | ||
| + | with normalization $G(1)=1$. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Corollary:</strong> The following values hold: | ||
| + | $$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.$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
Revision as of 00:34, 21 March 2015
The Barnes $G$ function is defined by the following Weierstrass factorization: $$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\},$$ where $\exp$ denotes the exponential function and $\gamma$ denotes the Euler-Mascheroni constant.
Properties
Theorem: The following formula holds: $$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: █
