Difference between revisions of "Manuals/calci/GFUNCTION"

Revision as of 16:47, 8 August 2017

GFUNCTION (Number)

This function shows the value of the Barnes G-function value.
In $GFUNCTION(Number)$ , $Number$ is any real number.
$G(z)$ is a function that is an extension of super factorials to the complex numbers.
It is related to the Gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes.
According to elementary factors, it is a special case of the double gamma function.
Formally, the Barnes G-function is defined in the following Weierstrass product form:

$G(1+z)={(2\pi )}^{\frac {z}{2}}exp(-{\frac {z+z^{2}(1+\gamma )}{2}})\prod _{k=1}^{\infty }[{(1+{\frac {z}{k}})}^{k}exp({\frac {z^{2}}{2k}}-z)]$

{{(1+\frac{z}{k})}^k

@@ Line 9: / Line 9: @@
 *According to elementary factors, it is a special case of the double gamma function.
 *Formally, the Barnes G-function is defined in the following Weierstrass product form:
-<math>G(1+z)={(2\pi)}^\frac{z}{2}exp(-\frac{z+z^2(1+\gamma)}{2})</math>
+<math>G(1+z)={(2\pi)}^\frac{z}{2}exp(-\frac{z+z^2(1+\gamma)}{2})\prod_{k=1}^\infty [{(1+\frac{z}{k})}^k exp(\frac {z^2}{2k}-z)]</math>
-*<math>\prod_{k=1}^\infty [{(1+\frac{z}{k})}^k exp(\frac {z^2}{2k}-z)]</math>
-<math> exp(\frac {z^2}{2k}-z)</math>
 {{(1+\frac{z}{k})}^k