Manuals/calci/GFUNCTION

GFUNCTION (Number)

• $Number$ is any positive real number.

Description

• 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:

$G(1+z)={(2\pi )}^{\frac {z}{2}}exp(-{\frac {z+z^{2}(1+\gamma )}{2}})$

• Failed to parse (syntax error): {\displaystyle \prod_{k=1}^\infty {{(1+\frac{z}{k})}^k }

Failed to parse (syntax error): {\displaystyle {{(1+\frac{z}{k})}^k exp(\frac {z^2}{2k}-z)}