Difference between revisions of "Glaisher–Kinkelin constant"
From specialfunctionswiki
(Created page with "The Glaisher–Kinkelin constant is defined by the formula $$A=\displaystyle\lim_{n \rightarrow \infty} \dfrac{(2\pi)^{\frac{n}{2}}n^{\frac{n^2}{2}-\frac{1}{12}}e^{-\frac{3n^2...") |
|||
| Line 1: | Line 1: | ||
The Glaisher–Kinkelin constant is defined by the formula | The Glaisher–Kinkelin constant is defined by the formula | ||
| − | $$A=\displaystyle\lim_{n \rightarrow \infty} \dfrac{(2\pi)^{\frac{n}{2}}n^{\frac{n^2}{2}-\frac{1}{12}}e^{-\frac{3n^2}{4}+\frac{1}{12}}}{G(n+1)} | + | $$A=\displaystyle\lim_{n \rightarrow \infty} \dfrac{(2\pi)^{\frac{n}{2}}n^{\frac{n^2}{2}-\frac{1}{12}}e^{-\frac{3n^2}{4}+\frac{1}{12}}}{G(n+1)},$$ |
| + | where $G$ is the [[Barnes G|Barnes $G$]] function. | ||
=Properties= | =Properties= | ||
{{:Derivative of zeta at -1}} | {{:Derivative of zeta at -1}} | ||
Revision as of 00:44, 21 March 2015
The Glaisher–Kinkelin constant is defined by the formula $$A=\displaystyle\lim_{n \rightarrow \infty} \dfrac{(2\pi)^{\frac{n}{2}}n^{\frac{n^2}{2}-\frac{1}{12}}e^{-\frac{3n^2}{4}+\frac{1}{12}}}{G(n+1)},$$ where $G$ is the Barnes $G$ function.
Contents
Properties
Theorem
The following formula holds: $$\zeta'(-1)=\dfrac{1}{12}-\log(A),$$ where $\zeta$ denotes the Riemann zeta function, $A$ denotes the Glaisher–Kinkelin constant, and $\log$ denotes the logarithm.
