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)]}
Examples
- GFUNCTION(10) = 5056584744960000
- GFUNCTION(4) = 2
- GFUNCTION(7) = 34560
- GFUNCTION(5.2) = 12