Manuals/calci/GFUNCTION

GFUNCTION (Number)

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Number} is any positive real number.

Description

• This function shows the value of the Barnes G-function value.
• In Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle GFUNCTION (Number)} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Number} is any real number.
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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