|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{for|the k-function|Bateman function}}
| | 47 year-old Zoologist Vernon Vancamp from High River, has interests such as painting, property developers in singapore and crocheting. Loves to discover unfamiliar cities and locales like Mont-Saint-Michel and its Bay.<br><br>my weblog: [http://Www.musicoda.com/entry.php?17446-Singapore-s-Main-Property-Portal-For-New-Condominium-Launches-( http://Www.musicoda.com] |
| In [[mathematics]], the '''K-function''', typically denoted ''K''(''z''), is a generalization of the [[hyperfactorial]] to [[complex number]]s, similar to the generalization of the [[factorial]] to the [[Gamma function]].
| |
| | |
| Formally, the K-function is defined as
| |
| | |
| :<math>K(z)=(2\pi)^{(-z+1)/2} \exp\left[\begin{pmatrix} z\\ 2\end{pmatrix}+\int_0^{z-1} \ln(t!)\,dt\right].</math>
| |
| | |
| It can also be given in closed form as
| |
| | |
| :<math>K(z)=\exp\left[\zeta^\prime(-1,z)-\zeta^\prime(-1)\right]</math>
| |
| | |
| where ζ'(''z'') denotes the [[derivative]] of the [[Riemann zeta function]], ζ(''a'',''z'') denotes the [[Hurwitz zeta function]] and
| |
| | |
| :<math>\zeta^\prime(a,z)\ \stackrel{\mathrm{def}}{=}\ \left[\frac{d\zeta(s,z)}{ds}\right]_{s=a}.</math>
| |
| | |
| Another expression using [[polygamma function]] is<ref>[http://www.cs.cmu.edu/~adamchik/articles/polyg.htm Victor S. Adamchik. PolyGamma Functions of Negative Order]</ref>
| |
| | |
| :<math>K(z)=\exp\left(\psi^{(-2)}(z)+\frac{z^2-z}{2}-\frac z2 \ln (2\pi)\right)</math>
| |
| | |
| Or using [[Generalized polygamma function|balanced generalization of Polygamma function]]:<ref>[http://www.math.tulane.edu/~vhm/papers_html/genoff.pdf Olivier Espinosa Victor H. Moll. A Generalized polygamma function. Integral Transforms and Special Functions Vol. 15, No. 2, April 2004, pp. 101–115]</ref>
| |
| | |
| :<math>K(z)=A e^{\psi(-2,z)+\frac{z^2-z}{2}}</math>
| |
| | |
| :where A is [[Glaisher constant]].
| |
| | |
| The K-function is closely related to the [[Gamma function]] and the [[Barnes G-function]]; for natural numbers ''n'', we have
| |
| | |
| :<math>K(n)=\frac{(\Gamma(n))^{n-1}}{G(n)}.</math>
| |
| | |
| More prosaically, one may write
| |
| | |
| :<math>K(n+1)=1^1\, 2^2\, 3^3 \cdots n^n.</math>
| |
| | |
| The first values are
| |
| :1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... ({{OEIS|A002109}}).
| |
| | |
| == References ==
| |
| | |
| <references />
| |
| | |
| ==External links==
| |
| * {{mathworld|title=K-Function|urlname=K-Function}}
| |
| | |
| [[Category:Gamma and related functions]]
| |
47 year-old Zoologist Vernon Vancamp from High River, has interests such as painting, property developers in singapore and crocheting. Loves to discover unfamiliar cities and locales like Mont-Saint-Michel and its Bay.
my weblog: http://Www.musicoda.com