Difference between revisions of "Barnes G"

From specialfunctionswiki
Jump to: navigation, search
 
(8 intermediate revisions by the same user not shown)
Line 2: Line 2:
 
$$G(1+z)=(2\pi)^{\frac{z}{2}} \exp \left( - \dfrac{z+z^2(1+\gamma)}{2} \right) \displaystyle\prod_{k=1}^{\infty} \left\{ \left( 1+\dfrac{z}{k} \right)^k \exp \left( \dfrac{z^2}{2k}-z \right) \right\},$$
 
$$G(1+z)=(2\pi)^{\frac{z}{2}} \exp \left( - \dfrac{z+z^2(1+\gamma)}{2} \right) \displaystyle\prod_{k=1}^{\infty} \left\{ \left( 1+\dfrac{z}{k} \right)^k \exp \left( \dfrac{z^2}{2k}-z \right) \right\},$$
 
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]].
 +
 +
<div align="center">
 +
<gallery>
 +
File:Barnesgplot.png|Graph of $G$.
 +
File:Complexbarnesgplot.png|[[Domain coloring]] of $G$.
 +
</gallery>
 +
</div>
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed">
+
[[Barnes G at z+1 in terms of Barnes G and gamma]]<br />
<strong>Theorem:</strong> The following formula holds:
+
[[Barnes G at positive integer]]<br />
$$G(z+1)=\Gamma(z)G(z)$$
+
 
with normalization $G(1)=1$.
+
=References=
<div class="mw-collapsible-content">
+
[http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN600494829_0031%7CLOG_0022 The theory of the $G$-function by E.W. Barnes]
<strong>Proof:</strong> █
 
</div>
 
</div>
 
  
<div class="toccolours mw-collapsible mw-collapsed">
+
[[Category:SpecialFunction]]
<strong>Corollary:</strong> The following values hold:
+
[[Category:Definition]]
$$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>
 

Latest revision as of 05:48, 6 June 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.

Properties

Barnes G at z+1 in terms of Barnes G and gamma
Barnes G at positive integer

References

The theory of the $G$-function by E.W. Barnes