Difference between revisions of "Manuals/calci/GFUNCTION"

Latest revision as of 16:02, 22 February 2019

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:

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

@@ 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 </math>
-<math>{{(1+\frac{z}{k})}^k exp(\frac {z^2}{2k}-z)</math>
+==Examples==
+# GFUNCTION(10) = 5056584744960000
+# GFUNCTION(4) = 2
+# GFUNCTION(7) = 34560
+# GFUNCTION(5.2) = 12
+==Related Videos==
+{{#ev:youtube|v=XZIVrkkYBRI&t=101s|280|center|Gamma Function}}
+==See Also==
+*[[Manuals/calci/BETAFUNCTION | BETAFUNCTION]]
+*[[Manuals/calci/KFUNCTION  | KFUNCTION ]]
+==References==
+*[https://en.wikipedia.org/wiki/Barnes_G-function  G Function]
+*[[Z_API_Functions | List of Main Z Functions]]
+*[[ Z3 |   Z3 home ]]