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Glaisher constant ：ウィキペディア英語版

Glaisher–Kinkelin constant
In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted ''A'', is a mathematical constant, related to the K-function and the Barnes G-function. The constant appears in a number of sums and integrals, especially those involving Gamma functions and zeta functions. It is named after mathematicians James Whitbread Lee Glaisher and Hermann Kinkelin.
Its approximate value is:
:

A\approx1.2824271291\dots

.
The Glaisher–Kinkelin constant

A

can be given by the limit:
:

A=\lim_ \frac}

where

K(n)=\prod_^ k^k

is the K-function. This formula displays a similarity between ''A'' and which is perhaps best illustrated by noting Stirling's formula:
:

\sqrt=\lim_ \frac}}

which shows that just as is obtained from approximation of the function

\prod_^ k

, ''A'' can also be obtained from a similar approximation to the function

\prod_^ k^k

.
An equivalent definition for ''A'' involving the Barnes G-function, given by

G(n)=\prod_^k!=\frac

where

\Gamma(n)

A=\lim_ \frac e^}

.
The Glaisher–Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function, such as:
:

\zeta^(-1)=\frac-\ln A

\sum_^\infty \frac=-\zeta^(2)=\frac\left(A-\gamma-\ln(2\pi)\right)

where

\gamma

is the Euler–Mascheroni constant. The latter formula leads directly to the following product found by Glaisher:
:

\prod_^ k^}}=\left(\frac}\right)^}

The following are some integrals that involve this constant:
:

\int_0^ \ln\Gamma(x)dx=\frac \ln A+\frac \ln 2+\frac \ln \pi

\int_0^\infty \fracdx=\frac \zeta^(-1)=\frac-\frac\ln A

A series representation for this constant follows from a series for the Riemann zeta function given by Helmut Hasse.
:

\ln A=\frac-\frac \sum_^\infty \frac \sum_^n \left(-1\right)^k \binom \left(k+1\right)^2 \ln(k+1)

==References==

*
* (Provides a variety of relationships.)
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抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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