Difference between revisions of "AbelFactorial"
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| + | #redirect [[AbelFactorial]] |
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| − | [[File:AbelFactorialR.png|500px|right|thumb|$y=\mathrm{AbelFactorial}(x)$ and $y=\mathrm{ArcFactorial}(x)$ versus $x$]] |
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| − | [[File:AbelFactorialMap.png|400px|right|thumb|Map of $f = \mathrm{AbelFactorial}(x+\mathrm i y)$ in the $x$,$y$ plane shown with isolines |
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| − | $p\!=\!\Re(f)\!=$const and |
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| − | $q\!=\!\Im(f)\!=$const]] |
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| − | '''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 |
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| − | : $\mathrm{AbelFactorial}(z^*)=\mathrm{AbelFactorial}(z)^*$. |
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| − | : $\mathrm{AbelFactorial}(3)=0$ |
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| − | : $\displaystyle |
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| − | \lim_{x\rightarrow -\infty}\mathrm{AbelFactorial}(x+\mathrm i y)=2 ~ ~ \forall y \in \mathbb R: y\!\ne\!0$ |
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| − | |||
| − | 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. |
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| − | |||
| − | 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 $\mathbb C \backslash \{ x\in \mathbb{R}: x\le 2\}$. |
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| − | ==Abel equation== |
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| − | AbelFactorial $G$ satisfies the [[Abel equation]] |
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| − | : (1) $~ ~ ~ G(z!)=G(z)+1$ |
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| − | The Abel equation is consistent with the [[transfer equation]] for the [[SuperFactorial]] |
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| − | $F=G^{-1}$: |
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| − | : (2) $~ ~ ~ \mathrm{Factorial}(F(z))=F(z\!+\!1)$ |
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| − | |||
| − | ==Regular iteration== |
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| − | 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"> |
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| − | http://tori.ils.uec.ac.jp/PAPERS/2010superfae.pdf |
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| − | D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12. |
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| − | <!-- (Russian version: p.8-14) |
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| − | http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 |
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| − | !--> |
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| − | </ref>: |
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| − | : (2) $~ ~ ~ |
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| − | \tilde G(z)= |
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| − | \frac{1}{k} \log\!\left(\sum_{n=1}^{N-1}U_{n}(z\!-\!2)^{n}+\mathcal{O}(z\!-\!2)^N\right) |
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| − | $ |
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| − | where |
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| − | : (3) $~ ~ ~ k=\ln\!\big(3+2\!~\mathrm{Factorial}'(0)\big)=\ln(3-2\!~\gamma) |
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| − | \approx 0.6127874523307$, |
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| − | <!-- |
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| − | 0836381366079016859252 |
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| − | !--> |
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| − | $\gamma$ is the [[Euler's constant]]; |
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| − | : $U_1 =1$ |
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| − | : $U_{2}=-\frac{\pi^2+6\gamma^{2}-18\gamma+6}{12(3-5\gamma+2\gamma^{2})} |
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| − | \approx 0.798731835$ |
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| − | |||
| − | <!-- %172434541585621072345730147 !--> |
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| − | 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 |
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| − | $z\!-\!2$. |
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| − | |||
| − | ==Extension of the regular iteration== |
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| − | If $|z\!-\!2|$ is not small, then the representation |
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| − | : (4) $~ ~ ~ \tilde G(z)=\tilde G (\mathrm{ArcFactorial}^n(z))+n$ |
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| − | can be be used for integer $n$. |
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| − | |||
| − | The AbelFactorial $G$ can be expressed through $\tilde{G}$; |
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| − | : (5) $~ ~ ~ G(z)=\tilde G(z) - \tilde G(3)$ |
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| − | in such a way that $G(3)=0$, corresponding to $\mathrm{SuperFactorial}(0)=3$. |
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| − | |||
| − | ==Previous notation== |
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| − | In publication <ref name="fac"> |
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| − | http://tori.ils.uec.ac.jp/PAPERS/2010superfae.pdf |
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| − | D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.</ref>, |
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| − | 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. |
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| − | |||
| − | ==Iteration of Factorial== |
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| − | |||
| − | Together with [[Superfactorial]] $F=G^{-1}$, the [[AbelFactorial]] $G$ allow to express the $c$-th iteration of [[Factorial]] as follows: |
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| − | : (6) $~ ~ ~ |
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| − | \mathrm{Factorial}^c(z)= F(c+G(z))$ |
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| − | where $c$ has no need to be integer. In particular,<br> |
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| − | at $c=1$, $~ \mathrm{Factorial}^c(z)=z!$;<br> |
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| − | at $c=0$, $~ \mathrm{Factorial}^c(z)=z$;<br> |
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| − | at $c=-1$, $~ \mathrm{Factorial}^c(z)=\mathrm{ArcFactorial}(z)$;<br> |
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| − | 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 |
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| − | <ref name="suomi2011"> |
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| − | http://tori.ils.uec.ac.jp/PDF/2011suomi.pdf |
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| − | D.Kouznetsov. NON-INTEGER ITERATES OF ANALYTIC FUNCTIONS. Slideshow presented at [[FMI]], Finland, 2011.06.15 |
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| − | </ref> |
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| − | : (7) $~ ~ ~ |
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| − | \sqrt{!\,} \Big( \sqrt{!\,}\big(z\big)\Big)= z! |
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| − | $ |
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| − | ==References== |
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| − | <references/> |
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| − | [[Category:Abel functions]] |
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| − | [[Category:Superfunctions]] |
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| − | [[Category:Articles in English]] |
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Latest revision as of 06:57, 1 December 2018
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