Английская Википедия:Glaisher–Kinkelin constant
In mathematics, the Glaisher–Kinkelin constant or Glaisher's constant, typically denoted Шаблон:Mvar, is a mathematical constant, related to the [[K-function|Шаблон:Mvar-function]] and the [[Barnes G-function|Barnes Шаблон:Mvar-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:
- Шаблон:Mvar = Шаблон:Val... Шаблон:OEIS.
The Glaisher–Kinkelin constant Шаблон:Mvar can be given by the limit:
- <math>A=\lim_{n\rightarrow\infty} \frac{H(n)}{n^{\frac{n^2}{2}+\frac{n}{2}+\frac{1}{12}}\,e^{-\frac{n^2}{4}}}</math>
where Шаблон:Math is the hyperfactorial. This formula displays a similarity between Шаблон:Mvar and Шаблон:Pi which is perhaps best illustrated by noting Stirling's formula:
- <math>\sqrt{2\pi}=\lim_{n \to \infty} \frac{n!}{n^{n+\frac12}\,e^{-n}}</math>
which shows that just as Шаблон:Pi is obtained from approximation of the factorials, Шаблон:Mvar can also be obtained from a similar approximation to the hyperfactorials.
An equivalent definition for Шаблон:Mvar involving the [[Barnes G-function|Barnes Шаблон:Mvar-function]], given by Шаблон:Math where Шаблон:Math is the gamma function is:
- <math>A=\lim_{n\rightarrow\infty} \frac{\left(2\pi\right)^\frac{n}{2} n^{\frac{n^2}{2}-\frac{1}{12}} e^{-\frac{3n^2}{4}+\frac{1}{12}}}{G(n+1)}</math>.
The Glaisher–Kinkelin constant also appears in evaluations of the derivatives of the Riemann zeta function, such as:
- <math>\zeta'(-1)=\tfrac{1}{12}-\ln A</math>
- <math>\sum_{k=2}^\infty \frac{\ln k}{k^2}=-\zeta'(2)=\frac{\pi^2} 6 \left( 12 \ln A - \gamma-\ln 2\pi \right)</math>
where Шаблон:Mvar is the Euler–Mascheroni constant. The latter formula leads directly to the following product found by Glaisher:
- <math>\prod_{k=1}^\infty k^\frac{1}{k^2} = \left(\frac{A^{12}}{2\pi e^\gamma} \right)^\frac{\pi^2}{6}</math>
An alternative product formula, defined over the prime numbers, reads [1]
- <math>\prod_{k=1}^\infty p_k^\frac{1}{p_k^2-1} = \frac{A^{12}}{2\pi e^\gamma}, </math>
where Шаблон:Math denotes the Шаблон:Mvarth prime number.
The following are some integrals that involve this constant:
- <math>\int_0^\frac12 \ln\Gamma(x) \, dx = \tfrac 3 2 \ln A+\frac 5 {24} \ln 2+\tfrac 1 4 \ln \pi</math>
- <math>\int_0^\infty \frac{x \ln x}{e^{2 \pi x}-1} \, dx = \tfrac 1 2 \zeta'(-1) = \tfrac 1 {24}-\tfrac 1 2 \ln A</math>
A series representation for this constant follows from a series for the Riemann zeta function given by Helmut Hasse.
- <math>\ln A=\tfrac 1 8 - \tfrac 1 2 \sum_{n=0}^\infty \frac 1 {n+1} \sum_{k=0}^n (-1)^k \binom n k (k+1)^2 \ln(k+1)</math>
References
- Шаблон:Cite journal
- Шаблон:Cite journal (Provides a variety of relationships.)
- Шаблон:Mathworld
- Шаблон:Mathworld
External links
