Difference between revisions of "SuperFactorial"
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| + | #redirect [[SuperFactorial]] |
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| − | [[File:Superfactorea500.png|right|300px|thumb|Fig.1. $y\!=\!x!$ and $y\!=\! \mathrm{SuperFactorial}(x)$ versus real $x$]] |
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| − | '''SuperFactorial''', or "superfactorial" is [[superfunction]] of [[factorial]] constructed at its [[fixed point]] 2. |
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| − | The smallest integer larger than 2 (id est 3) is chosen as its value at zero, $\mathrm{SuperFactorial}(0)=3~$. Then,<br> |
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| − | $\mathrm{SuperFactorial}(1)$ $=$ $3!$ $=$ $6$, <br> |
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| − | $\mathrm{SuperFactorial}(2)=6!=720$, <br> |
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| − | $\mathrm{SuperFactorial}(3)=720!~$ and so on;<br> |
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| − | : $\mathrm{SuperFactorial}(z)=\mathrm{Factorial}^z(3)$ |
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| − | Here, the upper index of a function indicates the number of [[iteration]]s. |
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| − | Superfactorial satisfies the [[transfer equation]] |
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| − | : $\mathrm{Factorial}(\mathrm{SuperFactorial}(z))=\mathrm{SuperFactorial}(z\!+\!1)$ |
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| − | SuperFactorial |
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| − | <ref name="fac"> |
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| − | http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.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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| − | </ref> |
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| − | should not be confused with [[hyperfactorial]] <ref name="hyperfac"> |
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| − | http://mathworld.wolfram.com/Hyperfactorial.html |
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| − | </ref>, expressed with |
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| − | : $\displaystyle \mathrm{Hyperfactorial}(n)=\prod_{k=1}^{n}k^k$ |
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| − | for integer values of the argument and with relatively simple integral |
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| − | for the complex values. |
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| − | ==Nest== |
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| − | Superfactorial can be expressed through the [[Mathematica]] built-in function [[Nest]], |
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| − | : $\mathrm{SuperFactorial}[z]=\mathrm{Nest}[\mathrm{Factorial}, 3, z]$ |
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| − | For $z\in \mathbb N$, such an expression is recognized correctly. |
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| − | <ref name="mathnest"> |
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| − | However, in the current (for year 2011) implementation of [[Nest]] function in the Mathematica software allows to evaluate the SuperFactorial only for very few values of the argument; in the most of cases, the attempt to use the Nest function for evaluation of SuperFactorial causes the only diagnostic message and cannot be used to plot the SuperFactorial.</ref> |
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| − | $\mathrm{SuperFactorial}(z)$ is just $z$th iteration of Fatorial, evaluated at 3; so, |
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| − | : $\mathrm{SuperFactorial}(0)=3$ |
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| − | : $\mathrm{SuperFactorial}(1)=\mathrm{Factorial}(\mathrm{SuperFactorial}(0))=\mathrm{Factorial}(3)=6$ |
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| − | : $\mathrm{SuperFactorial}(2)=\mathrm{Factorial}^2(3)=\mathrm{Factorial} |
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| − | \big( \mathrm{Factorial}(3)\big)=\mathrm{Factorial}(6)=6!=720$ |
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| − | : $\mathrm{SuperFactorial}(3)=\mathrm{Factorial}^3(3)=\mathrm{Factorial}(720)=720!$ |
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| − | : $\mathrm{SuperFactorial}(4)=\mathrm{Factorial}^4(3)=\mathrm{Factorial}(720!) ~$ |
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| − | and so on; however, the factorial of 720 is already too big and cannot be shown here as integer constant. Along the real axis, the SuperFactorial grows up faster than [[tetration]] does. |
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| − | |||
| − | ==History== |
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| − | First, the [[superfunction]] of factorial is described in 2010 in the [[Moscow University Physics Bulletin]] |
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| − | <ref name="fac"> |
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| − | http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.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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| − | </ref>. |
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| − | That publication defined the $\sqrt{!\,}$ as holomorphic function, |
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| − | giving sense to the logo of the Physics Department of that University. |
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| − | As a by–product, the [[SuperFunction]] of Factorial and that for some other special functions are considered there. |
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| − | Below, one of superfunctions of factorial is called SuperFactorial. |
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| − | <!-- |
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| − | Aiming the evaluation of $\sqrt{!\,}$, |
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| − | value SuperFactorial(0) is not adjusted there, it is used as it comes from the [[regular iteration]] while the coefficient with the leading exponential is just unity. |
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| − | Then the SuperFactorial appears as iteration of factorial, evaluated at about $3.8$; however, the integer value at zero is simpler to deal with. |
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| − | !--> |
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| − | The more detailed historic overview is presented in the article [[Square root of factorial]]. |
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| − | |||
| − | ==Regular iteration== |
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| − | The straightforward application of the [[regular iteration]] to the solution $F$ of the [[transfer equation]] |
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| − | :$ (1)~ ~ ~ ~ ~ F(z\!+\!1) = \mathrm{Factorial}(F(z))$ |
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| − | is described below. |
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| − | Aiming to build–up the growing function, the biggest among the real [[fixed point]]s of factorial should be chosen as value at $-\infty$, so, search the solution in the following form: |
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| − | :$(2)~ ~ ~ ~ ~ F(z)=2+ a_1 \varepsilon + a_2 \varepsilon^2 +..$ |
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| − | where |
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| − | :$(3)~ ~ ~ ~ ~ \varepsilon=\exp(kz)$ |
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| − | It is convenient to set $a_1=1~$. Then other coefficients $a$ and constant $k$ can be determined substituting such a representation into the transfer equation (1): |
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| − | :$(4) ~ ~ ~ ~ ~ \mathrm{Factoral}(2+ \varepsilon + a_2 \varepsilon^2 +..) - (2+ K \varepsilon + K^2 a_2 \varepsilon^2 +..)=0$ |
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| − | where $K=\exp(k)$. The expansion of the left hand side of equation (1) into the power series with respect to $\varepsilon$ and equalizing to zero the coefficients gives the equation for $K$ and those for $a_2$, $a_3$, .. For example, |
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| − | : $\mathrm{Factorial}'(2)-K=0$ |
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| − | : $\mathrm{Factorial}'(2) a_2+\mathrm{Factorial}''(2)\frac{1}{2} a_1^2-K^2 a_2=0$ |
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| − | The derivatives of Factorial are known, in particular, |
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| − | : $\mathrm{Factorial}'(2)=3-2 \gamma$, |
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| − | : $\mathrm{Factorial}''(2)= 2 - 6 \gamma + 2 \gamma^2 + \pi^2/3$ |
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| − | where $\gamma=-\mathrm{Factorial}'(0) \approx -0.5772156649015329$ is the [[Euler constant]]. |
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| − | Then, |
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| − | : $K=3-2\gamma~$ ; $~ ~ ~k=\ln(K)=\ln(3-2\!~\gamma) \approx 0.6127874523307$ <!--0836381366079008240243 !--> |
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| − | : $a_2=\frac{\pi^2+6\gamma^{2}-18\gamma+6}{12(3-5\gamma+2\gamma^{2})} \approx 0.79873183517243454$ <!--1585621072345730147!--> |
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| − | and similar expressions for higher coefficients $a$. |
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| − | The series (2) diverges, but the equation (1) allow to transfer the approximation from the range where it is valid (sufficient values of $-\Re(z)$) to other values and therefore get the required precision of evaluation of $F$. <!--with the asymptotic series.!--> |
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| − | In order to satisfy condition SuperFactorial(0)=3, it is sufficient to set |
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| − | : $\mathrm{SuperFactorial}(z)=F(x_0\!+\!z)$ |
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| − | where $~x_0\!\approx\! -0.91938596545217788~$ is solution of equation $~F(x_0)\!=\!3~$. |
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| − | ==Properties of Superfactorial== |
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| − | [[File:Superfactocomple1.png|right|400px| thumb|Fig.2. [[Complex map]] of $f=\mathrm{SuperFactorial}(x+\mathrm i y)$ in the $x,y$ plane; levels |
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| − | $u=\Re(f)=$const and |
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| − | $v=\Im(f)=$const are drawn]] |
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| − | The [[complex map]] of SuperFactorial is shown at the figure 2. |
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| − | <!-- |
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| − | at right. The [[contour plot]]s of<br> |
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| − | $ u\!=\!\Re(\mathrm {SuperFactorial} (x\!+\!\mathrm i y)$ and<br> |
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| − | $ v\!=\!\Im(\mathrm {SuperFactorial} (x\!+\!\mathrm i y)$<br> |
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| − | intersect on the right angles, as they are supposed to do at the complex map of any [[holomorphic function]]. |
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| − | The integer values of $u$ or $v$ are marked with thick lines; thin lines correspond to the intermediate levels. |
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| − | !--> |
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| − | SuperFactorial is [[real-holomorphic]], at lest in the range $\{z \in \mathbb C : \Re(z)\le 2\}$. |
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| − | <!-- |
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| − | $\mathrm{SuperFactorial}(z^*)= \mathrm{SuperFactorial}(z)^*$ |
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| − | for all $z$ from the range of the definition. |
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| − | !--> |
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| − | SuperFactorial is [[periodic function]]; the period |
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| − | : $T=\frac{2 \pi \mathrm i}{k} \approx 10.2534496811560279265772640691397~\mathrm i$ |
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| − | At non-negative integer values of the argument, SuperFactorial takes integer values; in particular, |
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| − | $\mathrm{SuperFactorial}(0)=3~$ |
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| − | Function $\tilde F(z)=\mathrm{SuperFactorial}\left(z+\frac{T}{2}\right)$ is also [[real–holomorphic]]; it smoothly decreases from value 2 at $-\infty$ to $1$ at $\infty$ along the real axis. Such a complementary superfunction typically exists at the consruction of a superfunction with [[regular iteration]] at a real [[fixed point]]. |
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| − | ==Singularities== |
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| − | [[File:Sfaczoo300.png|right|500px|thumb|Fig.3. Zoom-in from Fig.2. |
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| − | <!--complex map of SuperFactorial, $f\!=\!\mathrm{SuperFactorial}(x\!+\!\mathrm i y)$; levels |
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| − | $u\!=\!\Re(f)=$const and |
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| − | $v\!=\!\Im(f)=$const are drawn!-->]] |
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| − | FuperFactorial is not [[entire function]], over-vice, it would have negative integer values somewhere, which is not consistent with the transfer equation (1): Factorial at negative integers has poles. The unity translation form any point, where the SuperFactorial has negative integer value leads to a singularity. |
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| − | The conclusion about the singularities is confirmed by the zoom-in from the comlpex map of the SuperFactorial. The figure indicates that the singularities of SuperFactorial are located within half-strips |
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| − | : $ z\in \mathbb C :$ $ \{ \Re(z) >\!2.7 ~,~ |\Im(z)\!+\!T n|<1 ~\mathrm{for~ some~} n\in \mathbb Z \}$ |
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| − | Outside these half-strips, the SuperFactorial exponentially approaches 2 as the real part of the argument goes to $-\!\infty$ and approaches unity as |
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| − | the real part of the argument goes to $+\!\infty$. |
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| − | ==AbelFactorial== |
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| − | The inverse function of SuperFactorial is [[AbelFactorial]]; at least in some vicinity of the [[half-line]] along the real axis, the relations |
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| − | : $\mathrm{SuperFactorial}( |
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| − | \mathrm{AbelFactorial}(z))=z$ |
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| − | : $\mathrm{AbelFactorial}( |
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| − | \mathrm{SuperFactorial}(z))=z$ |
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| − | The AbelFactorial satisfies the [[Abel equation]] |
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| − | : $\mathrm{Abelfactorial}( \mathrm{Factorial}(z))= \mathrm{AbelFactorial}(z)+1$ |
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| − | See the special article [[AbelFactorial]] for the details. |
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| − | Together, the AbelFactorial and the SuperFactorial allow to express (and to evaluate) the iterations of Factorial in the following way: |
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| − | : $\mathrm{Factorial}^c(z)=\mathrm{SuperFactorial}\big(c + \mathrm{AbelFactorial}(z)\big)$ |
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| − | where number $c$ of iterations has no need to be integer; it can be even complex, and for some domain of $c,d,z\in C^3 \subseteq \mathbb C^3$, the relation |
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| − | : $\mathrm{Factorial}^c\big(\mathrm{Factorial}^d(z)\big)=\mathrm{Factorial}^{c+d}((z)$ |
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| − | as if takes place for the exponentiation. This justifies the notation [[square root of factorial]] for the half–iteration of Factorial, id est |
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| − | : $\sqrt{!~}~=~\mathrm{Factorial}^{1/2}$ |
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| − | In wide domain of values of $z$, the relation |
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| − | : $\sqrt{!~}~(\sqrt{!~}(z))~=~\mathrm{Factorial}(z)=z!$ |
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| − | See the special article [[square root of factorial]] for the details. |
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| − | ==Existence and uniqueness== |
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| − | The existence and uniquness of a SuperFunction for a holomorphic fixed point is considered in |
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| − | <ref name="sqrt2"> |
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| − | http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html |
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| − | D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756. |
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| − | </ref>. The regular iteration gives the explicit proof of existence, while the requirements on the behavior at zero and the asymptotic properties provide the uniqueness. |
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| − | However, there are many other fixed points, and each of them can be used to construct a [[superfunction]] of [[factorial]], pretty different from the SuperFactorial described above. |
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| − | The new superfunctions of Factorial can be constructed with periodic perturbation of its argument, let |
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| − | : $f(z)=\mathrm{SuperFactorial}(z+\zeta(z))$ |
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| − | where $\zeta$ is some hoomorphic function, periodic with period unity. |
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| − | Function $f$ satisfies the same transfer equation {1} as the SuperFactorial; |
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| − | however, the periodicity or $\zeta$ does not allow it to decay as the real part of the argument goes $-\infty$; |
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| − | as the result, neither the asymptotic behavior (2) of the SuperFactorial, nor its periodicity with period $T$ can be achieved. |
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| − | In this sense, with specification of the asymptotic behavior and the periodicity, the SuperFactorial is unique. |
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| − | The SuperFactorial above seems to be the simplest among various SuperFunctions of Factorial, if one exclude functions that are identically equal to a constant. |
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| − | Other non-trivial superfunctions of Factorial show complicated behavior: and even the holomorphism in the left hand side of the complex plane cannot be achieved. Up to year 2011, a statement has status of [[conjecture]], and the rigorous proof may be matter for the good [[outstanding research]]. |
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| − | ==Algorithmic implementation of Superfactorial== |
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| − | ===C++ implementation=== |
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| − | For the [[efficient evaluation]] of SuperFactorial, the [[regular iteration]] is suitable; the reasonable number of terms in the partial sum is of order of dozen. |
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| − | Then, within few tens of operations, one can get of order of 15 decimal digits, that is sufficient for the complex double implementation. |
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| − | The C++ code for the evaluation of [[SuperFactorial]] used to generate the figures is expected to be loaded, as a common component of the generators of the figures. |
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| − | In the final version, the SuperFactorial is expected to be implemented as |
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| − | : complex<double> SuperFactorial(complex<double> z) |
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| − | and its inverse function is expected to be implemented as |
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| − | : complex<double> AbelFactorial(complex<double> z) |
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| − | The preliminary versions of the routines can be uploaded upon request; the final version will appear after to check the compatibility with other codes that are already loaded in TORI and in the [[Citizendium]]. |
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| − | ===Mathematica implementation=== |
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| − | In version of 2011, [[Mathematica]] already has name reserved for the iteration of functions, namely, [[Nest (Mathematica)|Nest]]. |
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| − | The call of such a function may have form |
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| − | :'''Nest[f,a,z]''' |
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| − | where $f$ indicates the name of the function, $z$ is number of iterations, and $a$ is initial value. |
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| − | Then, $\mathrm{SuperFactorial}(z)$ can be expressed with |
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| − | : '''Nest[Factorial,3,z]''' |
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| − | and the $c$th iteration of [[Factorial]], id est, $\mathrm{Factorial}^c(z)$ could be written as |
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| − | : '''Nest[Factorial,z,c]''' |
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| − | In the current version of Mathematica (year 2011), the '''Nest''' is implemented only for the case, when the last argument can be simplified to an constant , expressed with a natural number. |
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| − | <ref name="mathematica"> |
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| − | http://reference.wolfram.com/mathematica/ref/Nest.html |
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| − | BUILT-IN MATHEMATICA SYMBOL Tutorials. Nest. |
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| − | </ref> |
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| − | The upgrade of the implementation of Nest may include the [[table of SuperFunctions]] and corresponding [[Abel function]]s, |
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| − | their properties and ways of the efficient evaluation with required precision. |
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| − | ==Conclusions== |
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| − | From the point of view of construction of [[superfunction]]s, there is nothing specific in the Factorial: the SuperFactorial is constructed in the same way as superfunction for $\exp_b$ for $1<b<\exp(1/\mathrm e)$, described in <ref name="sqrt2">sqrt2</ref> or that for the [[logistic operator]] |
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| − | <ref name="logistic"> |
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| − | http://www.springerlink.com/content/u712vtp4122544x4 |
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| − | D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98. |
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| − | </ref>. |
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| − | SupeFactorial, together with its inverse function (id est, [[AbelFactorial]]) allows to define the $c$th iteration of factorial for non-integer values of $c$; in particilar, the [[square root of factorial]] (that was believed to have no sense |
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| − | <ref name="kandidov"> |
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| − | http://ofvp.phys.msu.ru/pdf/Kandidov_70.pdf: |
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| − | V.P.Kandidov. About the time and myself. (In Russian) |
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| − | <blockquote>По итогам студенческого голосования победителями оказались значок с изображением |
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| − | рычага, поднимающего Землю, и нынешний с хорошо известной эмблемой в виде |
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| − | корня из факториала, вписанными в букву Ф. Этот значок, созданный студентом |
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| − | кафедры биофизики А.Сарвазяном, привлекал своей простотой и |
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| − | выразительностью. Тогда эмблема этого значка подверглась жесткой критике со |
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| − | стороны руководства факультета, поскольку она не имеет физического смысла, |
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| − | математически абсурдна и идеологически бессодержательна. |
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| − | </blockquote> |
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| − | </ref><ref name="naukai"> |
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| − | http://nauka.relis.ru/11/0412/11412002.htm |
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| − | 250 anniversary of the Moscow State University. (In Russian) |
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| − | ПЕРВОМУ УНИВЕРСИТЕТУ СТРАНЫ - 250! |
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| − | <blockquote> |
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| − | На значке физфака в букву "Ф" вписано стилизованное изображение корня из факториала (√!) - выражение, математического смысла не имеющее. |
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| − | </blockquote> |
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| − | </ref>). appears as half-iteratiopn of factorial. This symbol of this function, id est, $\sqrt{!\,}$ is used as central |
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| − | part of logo of [[TORI]]. |
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| − | ==References== |
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| − | <references/> |
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| − | |||
| − | [[Category:Superfunction]] |
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| − | [[Category:SuperFactorial]] |
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| − | [[Category:Factorial]] |
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| − | [[Category:Holomorphic function]] |
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| − | [[Category:Articles in English]] |
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Revision as of 07:06, 1 December 2018
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