Difference between revisions of "AbelFactorial"

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#redirect [[AbelFactorial]]
[[File:AbelFactorialR.png|500px|right|thumb|$y=\mathrm{AbelFactorial}(x)$ and $y=\mathrm{ArcFactorial}(x)$ versus $x$]]
 
[[File:AbelFactorialMap.png|400px|right|thumb|Map of $f = \mathrm{AbelFactorial}(x+\mathrm i y)$ in the $x$,$y$ plane shown with isolines
 
$p\!=\!\Re(f)\!=$const and
 
$q\!=\!\Im(f)\!=$const]]
 
'''AbelFactorial''' is holomorphic function, inverse of the [[SuperFactorial]], [[Abel function]] of [[Factorial]] constructed with [[regular iteration]] at its fixed points 2 in such a way that
 
: $\mathrm{AbelFactorial}(z^*)=\mathrm{AbelFactorial}(z)^*$.
 
: $\mathrm{AbelFactorial}(3)=0$
 
: $\displaystyle
 
\lim_{x\rightarrow -\infty}\mathrm{AbelFactorial}(x+\mathrm i y)=2 ~ ~ \forall y \in \mathbb R: y\!\ne\!0$
 
 
Along the real axis, AbelFactorial is slowly growing function, its growth is much slower than that of [[ArcFactorial]] shown in the top figure at right.
 
 
ArcFactorial is analytic in the complex plane with cut in the direction of the negative part f the real axis; the range of [[holomorphizm]] is $\mathbb C \backslash \{ x\in \mathbb{R}: x\le 2\}$.
 
==Abel equation==
 
AbelFactorial $G$ satisfies the [[Abel equation]]
 
: (1) $~ ~ ~ G(z!)=G(z)+1$
 
The Abel equation is consistent with the [[transfer equation]] for the [[SuperFactorial]]
 
$F=G^{-1}$:
 
: (2) $~ ~ ~ \mathrm{Factorial}(F(z))=F(z\!+\!1)$
 
 
==Regular iteration==
 
For the evaluation of AbelFactorial, the [[regular iteration]] at the fixed point 2 of factorial is efficient. The following expansion is suggested <ref name="fac">
 
http://tori.ils.uec.ac.jp/PAPERS/2010superfae.pdf
 
D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
 
<!-- (Russian version: p.8-14)
 
http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1
 
!-->
 
</ref>:
 
: (2) $~ ~ ~
 
\tilde G(z)=
 
\frac{1}{k} \log\!\left(\sum_{n=1}^{N-1}U_{n}(z\!-\!2)^{n}+\mathcal{O}(z\!-\!2)^N\right)
 
$
 
where
 
: (3) $~ ~ ~ k=\ln\!\big(3+2\!~\mathrm{Factorial}'(0)\big)=\ln(3-2\!~\gamma)
 
\approx 0.6127874523307$,
 
<!--
 
0836381366079016859252
 
!-->
 
$\gamma$ is the [[Euler's constant]];
 
: $U_1 =1$
 
: $U_{2}=-\frac{\pi^2+6\gamma^{2}-18\gamma+6}{12(3-5\gamma+2\gamma^{2})}
 
\approx 0.798731835$
 
 
<!-- %172434541585621072345730147 !-->
 
The coefficients $U$ can be found substituting the representation (2) into the Abel equation (1) and expanding the result to the power series with small parameter
 
$z\!-\!2$.
 
 
==Extension of the regular iteration==
 
If $|z\!-\!2|$ is not small, then the representation
 
: (4) $~ ~ ~ \tilde G(z)=\tilde G (\mathrm{ArcFactorial}^n(z))+n$
 
can be be used for integer $n$.
 
 
The AbelFactorial $G$ can be expressed through $\tilde{G}$;
 
: (5) $~ ~ ~ G(z)=\tilde G(z) - \tilde G(3)$
 
in such a way that $G(3)=0$, corresponding to $\mathrm{SuperFactorial}(0)=3$.
 
 
==Previous notation==
 
In publication <ref name="fac">
 
http://tori.ils.uec.ac.jp/PAPERS/2010superfae.pdf
 
D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.</ref>,
 
function $\tilde G$ is called "ArcSuperFactorial" and denoted with letter $G$; however, it is more convenient to deal with a function that takes integer values at least for some integer values of the argument; therefore the representation (5) is recommended.
 
 
==Iteration of Factorial==
 
 
Together with [[Superfactorial]] $F=G^{-1}$, the [[AbelFactorial]] $G$ allow to express the $c$-th iteration of [[Factorial]] as follows:
 
: (6) $~ ~ ~
 
\mathrm{Factorial}^c(z)= F(c+G(z))$
 
where $c$ has no need to be integer. In particular,<br>
 
at $c=1$, $~ \mathrm{Factorial}^c(z)=z!$;<br>
 
at $c=0$, $~ \mathrm{Factorial}^c(z)=z$;<br>
 
at $c=-1$, $~ \mathrm{Factorial}^c(z)=\mathrm{ArcFactorial}(z)$;<br>
 
and at $c=1/2$, such a representation determines the square root of factorial, id est, $\sqrt{!\,}$, that is used as Logo of the Physics Department of the Moscow State University and part of logo of [[TORI]] shown in the left top corner of each page of TORI. In some facility of the half-line $z\!>\!2$, the following relation holds
 
<ref name="suomi2011">
 
http://tori.ils.uec.ac.jp/PDF/2011suomi.pdf
 
D.Kouznetsov. NON-INTEGER ITERATES OF ANALYTIC FUNCTIONS. Slideshow presented at [[FMI]], Finland, 2011.06.15
 
</ref>
 
: (7) $~ ~ ~
 
\sqrt{!\,} \Big( \sqrt{!\,}\big(z\big)\Big)= z!
 
$
 
 
==References==
 
<references/>
 
 
[[Category:Abel functions]]
 
[[Category:Superfunctions]]
 
[[Category:Articles in English]]
 

Latest revision as of 06:57, 1 December 2018

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