Difference between revisions of "File:Sfaczoo300.png"

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{{oq|Sfaczoo300.png|Original file ‎(1,142 × 453 pixels, file size: 302 KB, MIME type: image/png)}}
−
Zoom-in of the [[complex map]] of [[SuperFactorial]]
  
   
  +
Zoom-in of the [[complex map]] of [[SuperFactorial]].
−
Copyleft 1011 by Dmitrii Kouznetsov
  
   
  +
[[SuperFactorial]] is described in the [[Moscow University Physics Bulletin]], 2010
−
[[Category:SuperFactorial]]
  
  +
<ref>
  +
http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1<br>
  +
http://mizugadro.mydns.jp/PAPERS/2010superfae.pdf reprint, English version<br>
  +
http://mizugadro.mydns.jp/PAPERS/2010superfar.pdf reprint, Russian version<br>
  +
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)
  +
</ref>.
  +
  +
Copyleft 2010 by Dmitrii Kouznetsov
  +
  +
This map is used as bottom part of Fig.8.6 at page 96 of book
  +
«[[Superfunctions]]»<ref name="bookA">
  +
https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas,algorithms,tables,graphics - [[Lambert Academic Publishing]], 2020/7/28
  +
</ref><ref name="bookM">https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas, algorithms, tables, graphics. Publisher: [[Lambert Academic Publishing]].
  +
</ref><br>
  +
in order to show the complicated structure of [[Superfactorial]]
  +
in the strip along the positive part of real axis (and in the translations for integer factor of the the period of [[Superfactorial]]).
  +
  +
==C++ generator of curves==
  +
  +
Files
  +
[[afacc.cin]],
  +
[[facp.cin]],
  +
[[superfac.cin]],
  +
[[ado.cin]],
  +
[[conto.cin]]
  +
should be loaded to the working directory in order to compile the [[C++]] code below:
  +
<pre>
  +
#include <math.h>
  +
#include <stdio.h>
  +
#include <stdlib.h>
  +
#define DB double
  +
#define DO(x,y) for(x=0;x<y;x++)
  +
using namespace std;
  +
#include <complex>
  +
typedef complex<double> z_type;
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
#include "fac.cin"
  +
//#include "sinc.cin"
  +
#include "facp.cin"
  +
#include "afacc.cin"
  +
#include "superfac.cin"
  +
#include "conto.cin"
  +
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  +
int M=551,M1=M+1;
  +
int N=221,N1=N+1;
  +
DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
  +
char v[M1*N1]; // v is working array
  +
FILE *o;o=fopen("SuperFacZoom.eps","w");ado(o,552,222);
  +
fprintf(o,"1 1 translate\n 100 100 scale\n");
  +
DO(m,M1)X[m]=.01*(m-.5);
  +
DO(n,N1)Y[n]=.01*(n-.5);
  +
  +
for(m=0;m<6;m++){M(m,0)L(m,2)}
  +
for(n=0;n<3;n++){M( 0,n)L(5,n)}
  +
fprintf(o,".004 W 0 0 0 RGB S\n");
  +
DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
  +
DO(m,M1){x=X[m]; //printf("%5.2f\n",x);
  +
DO(n,N1){y=Y[n]; z=z_type(x,y);
  +
// c=afacc(z);
  +
// c=fac(z);
  +
c=superfac(z);
  +
// p=abs(c-d)/(abs(c)+abs(d)); p=-log(p)/log(10.)-1.;
  +
p=Re(c);q=Im(c);
  +
if(p>-20 && p<20 &&
  +
// (fabs(y)>.034 ||x>-.9 ||fabs(x-int(x))>1.e-3) &&
  +
q>-20 && q<20 && fabs(q)> 1.e-16
  +
)
  +
{g[m*N1+n]=p;f[m*N1+n]=q;}
  +
}}
  +
//fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1.8;q=.7;
  +
  +
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1.4;q=.8;
  +
for(m=-4;m<4;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".004 W 0 .5 0 RGB S\n");
  +
for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".004 W .8 0 0 RGB S\n");
  +
for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".004 W 0 0 .8 RGB S\n");
  +
for(m=1;m<15;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".01 W .8 0 0 RGB S\n");
  +
for(m=1;m<15;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".01 W 0 0 .8 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-9,9); fprintf(o,".01 W .5 0 .5 RGB S\n");
  +
for(m=-14;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  +
m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-9,9); fprintf(o,".01 W 0 0 0 RGB S\n");
  +
for(m=1;m<17;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
  +
//#include"plofu.cin"
  +
// x=0.8856031944;
  +
conto(o,g,w,v,X,Y,M,N,0.8856031944,-p,p); fprintf(o,".004 W .2 .2 0 RGB S\n");
  +
/*
  +
M(x,-8)L(x,8) fprintf(o,"0 setlinejoin 0 setlinecap 0.004 W 0 0 0 RGB S\n");
  +
M(x,0)L(-8.1,0) fprintf(o," .05 W 1 1 1 RGB S\n");
  +
DO(m,23){ M(x-.4*m,0)L(x-.4*(m+.5),0);} fprintf(o,".09 W .3 .3 0 RGB S\n");
  +
//M(x,0)L(-8.1,0) fprintf(o,"[.19 .21]0 setdash .05 W 0 0 0 RGB S\n");
  +
// May it be, that, some printers do not interpret well the dashing ?
  +
*/
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf SuperFacZoom.eps");
  +
system( "open SuperFacZoom.pdf"); //for LINUX
  +
// getchar(); system("killall Preview");//for mac
  +
}
  +
</pre>
  +
  +
==Generator of labels==
  +
<pre>
  +
\documentclass[12pt]{article}
  +
\usepackage{geometry}
  +
\usepackage{graphicx}
  +
\usepackage{rotating}
  +
\newcommand{\sx}{\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing \includegraphics
  +
\pagestyle{empty}
  +
\topmargin -82pt
  +
\oddsidemargin -97pt
  +
\textwidth 1600pt
  +
\textheight 1600pt
  +
\paperwidth 1110pt
  +
\paperheight 452pt
  +
  +
\begin{document}
  +
%\sx{1.2}{\begin{picture}(346,346) \ax
  +
\sx{2}{ \begin{picture}(362,212)
  +
%\sx{.88}{ \begin{picture}(362,210)
  +
\put(0,0){\ing{SuperFacZoom}}
  +
\put(4,216){\sx{1.8}{$y$}}
  +
\put(4,194){\sx{1.8}{$2$}}
  +
\put(4,095){\sx{1.8}{$1$}}
  +
\put(097,3){\sx{1.8}{$1$}}
  +
\put(197,3){\sx{1.8}{$2$}}
  +
\put(297,3){\sx{1.8}{$3$}}
  +
\put(397,3){\sx{1.8}{$4$}}
  +
\put(497,3){\sx{1.8}{$5$}}
  +
\put(536,3){\sx{1.8}{$x$}}
  +
  +
\put(360,210){\sx{1.8}{$u\!=\!0.8856031944$}}
  +
\put(311,198){\sx{1.3}{\rot{-51}$u\!=\!0.8$\ero}}
  +
\put(146,159){\sx{2}{ \rot{62}$u\!=\!1$\ero }}
  +
\put(172,117){\sx{2}{ \rot{1}$u\!=\!0$\ero }}
  +
\put(020,163){\sx{2}{ \rot{-37}$u\!=\!2$\ero }}
  +
\put(018,053){\sx{2}{ \rot{ 22}$u\!=\!3$\ero }}
  +
\put(051,06){\sx{2}{ \rot{ 53}$u\!=\!4$\ero }}
  +
  +
\put(082,183){\sx{2}{ \rot{2}$v\!=\!1$\ero }}
  +
\put(081,083){\sx{2}{ \rot{46}$v\!=\!2$\ero }}
  +
\put(306,148){\sx{2}{ \rot{0.}$v\!=\!0$\ero }}
  +
\put(406,148){\sx{2}{ \rot{0.}$v\!=\!0$\ero }}
  +
\end{picture}
  +
}
  +
\end{document}
  +
</pre>
  +
==Warning==
  +
  +
While only one [[Superfactorial]] is described,
  +
terms
  +
«[[Super Factorial]]»,
  +
«[[SuperFactorial]]»,
  +
«[[Superfactorial]]»,
  +
«[[SuFac]]»<br>
  +
are treated as synonyms denoting the [[Superfunction]] of [[Factorial]]<br>
  +
constructed with [[Regular iteration]] at the fixed point \(L\!=\!2\)<br>
  +
that grows infinitely along the real axis and<br>
  +
has value 3 (minimal integer bigger than this \(L\)) at zero.
  +
  +
Factorial has many fixed points, and two of them are positive.
  +
  +
As soon as another [[Superfactorial]] be described,
  +
the notations need to be adjusted. Suggestion:
  +
  +
[[SuFac]] for that shown above, and [[AuFac]] for its inverse,<br>
  +
[[SdFac]] for that at the fixed point \(L\!=\!1\), and [[AdFac]] for its inverse,<br>
  +
[[SuperFac]]\(_n\) for another, constructed at the \(n\)th fixed point \(L_n\) of Factorial.
  +
  +
Let the colleague(s) who describes other super factorials (perhaps without fast growth along the real axis) <br>
  +
chose(s) the appropriate names top denote them.
  +
  +
==References==
  +
{{ref}}
  +
{{fer}}
  +
  +
==Keywords==
  +
«[[]]»,
  +
<b>«[[Factorial]]»</b>,
  +
«[[Iterate]]»,
  +
«[[Regular iteration]]»,
  +
<b>«[[SuFac]]»</b>,
  +
«[[SuperFactorial]]»,
  +
«[[Superfactorial]]»,
  +
«[[Superfunction]]»,
  +
«[[Superfunctions]]»,
  +
«[[]]»,
  +
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:C++]]
 
[[Category:Complex maps]]
  
[[Category:Complex maps]]
  +
[[Category:Latex]]
  +
[[Category:Regular iteration]]
  +
[[Category:SuFac]]
  +
[[Category:SuperFactorial]]
  +
[[Category:Superfactorial]]
  +
[[Category:SuperFunction]]
 
[[Category:SuperFunctions]]
  
[[Category:SuperFunctions]]

Latest revision as of 23:00, 22 August 2025


Zoom-in of the complex map of SuperFactorial.

SuperFactorial is described in the Moscow University Physics Bulletin, 2010 [1].

Copyleft 2010 by Dmitrii Kouznetsov

This map is used as bottom part of Fig.8.6 at page 96 of book «Superfunctions»[2][3]
in order to show the complicated structure of Superfactorial in the strip along the positive part of real axis (and in the translations for integer factor of the the period of Superfactorial).

C++ generator of curves

Files afacc.cin, facp.cin, superfac.cin, ado.cin, conto.cin should be loaded to the working directory in order to compile the C++ code below: 

 #include <math.h>
 #include <stdio.h>
 #include <stdlib.h>
 #define DB double
 #define DO(x,y) for(x=0;x<y;x++)
 using namespace std;
 #include <complex>
 typedef complex<double> z_type;
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)
 #include "fac.cin"
 //#include "sinc.cin"
 #include "facp.cin"
 #include "afacc.cin"
 #include "superfac.cin"
 #include "conto.cin"

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
   int M=551,M1=M+1;
   int N=221,N1=N+1;
 DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
 char v[M1*N1]; // v is working array
 FILE *o;o=fopen("SuperFacZoom.eps","w");ado(o,552,222);
 fprintf(o,"1 1 translate\n 100 100 scale\n");
DO(m,M1)X[m]=.01*(m-.5);
DO(n,N1)Y[n]=.01*(n-.5);

 for(m=0;m<6;m++){M(m,0)L(m,2)}
 for(n=0;n<3;n++){M(  0,n)L(5,n)}
 fprintf(o,".004 W 0 0 0 RGB S\n");
 DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
 DO(m,M1){x=X[m]; //printf("%5.2f\n",x);
 DO(n,N1){y=Y[n]; z=z_type(x,y);         
 //     c=afacc(z);
 //     c=fac(z);
        c=superfac(z);
 //     p=abs(c-d)/(abs(c)+abs(d));  p=-log(p)/log(10.)-1.;
        p=Re(c);q=Im(c);        
        if(p>-20 && p<20 &&
 //       (fabs(y)>.034 ||x>-.9 ||fabs(x-int(x))>1.e-3) &&
           q>-20 && q<20 && fabs(q)> 1.e-16
        ) 
        {g[m*N1+n]=p;f[m*N1+n]=q;}
                        }}
 //fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1.8;q=.7;

 fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1.4;q=.8;
 for(m=-4;m<4;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".004 W 0 .5 0 RGB S\n");
 for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".004 W .8 0 0 RGB S\n");
 for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".004 W 0 0 .8 RGB S\n");
 for(m=1;m<15;m++)  conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".01 W .8 0 0 RGB S\n");
 for(m=1;m<15;m++)  conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".01 W 0 0 .8 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (0.  ),-9,9); fprintf(o,".01 W .5 0 .5 RGB S\n");
 for(m=-14;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
           m=0;     conto(o,g,w,v,X,Y,M,N, (0.+m),-9,9); fprintf(o,".01 W 0 0 0 RGB S\n");
 for(m=1;m<17;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".01 W 0 0 0 RGB S\n");
 //#include"plofu.cin"
 // x=0.8856031944;
  conto(o,g,w,v,X,Y,M,N,0.8856031944,-p,p); fprintf(o,".004 W .2 .2 0 RGB S\n");
/*
 M(x,-8)L(x,8) fprintf(o,"0 setlinejoin 0 setlinecap 0.004 W 0 0 0 RGB S\n");
 M(x,0)L(-8.1,0) fprintf(o,"                   .05 W  1 1 1 RGB S\n");
 DO(m,23){ M(x-.4*m,0)L(x-.4*(m+.5),0);} fprintf(o,".09 W  .3 .3 0 RGB S\n");
 //M(x,0)L(-8.1,0) fprintf(o,"[.19 .21]0 setdash .05 W  0 0 0 RGB S\n");
 // May it be, that, some printers do not interpret well the dashing ?
*/
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf SuperFacZoom.eps");    
        system(    "open SuperFacZoom.pdf");    //for LINUX
 //     getchar(); system("killall Preview");//for mac
 }

Generator of labels

\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphicx}
\usepackage{rotating}
\newcommand{\sx}{\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing \includegraphics
\pagestyle{empty}
\topmargin -82pt
\oddsidemargin -97pt
\textwidth 1600pt
\textheight 1600pt
\paperwidth 1110pt
\paperheight 452pt

\begin{document}
%\sx{1.2}{\begin{picture}(346,346) \ax
 \sx{2}{ \begin{picture}(362,212)
 %\sx{.88}{ \begin{picture}(362,210)
 \put(0,0){\ing{SuperFacZoom}}
\put(4,216){\sx{1.8}{$y$}}
\put(4,194){\sx{1.8}{$2$}}
\put(4,095){\sx{1.8}{$1$}}
\put(097,3){\sx{1.8}{$1$}}
\put(197,3){\sx{1.8}{$2$}}
\put(297,3){\sx{1.8}{$3$}}
\put(397,3){\sx{1.8}{$4$}}
\put(497,3){\sx{1.8}{$5$}}
\put(536,3){\sx{1.8}{$x$}}

\put(360,210){\sx{1.8}{$u\!=\!0.8856031944$}}
\put(311,198){\sx{1.3}{\rot{-51}$u\!=\!0.8$\ero}}
\put(146,159){\sx{2}{ \rot{62}$u\!=\!1$\ero }}
\put(172,117){\sx{2}{ \rot{1}$u\!=\!0$\ero }}
\put(020,163){\sx{2}{ \rot{-37}$u\!=\!2$\ero }}
\put(018,053){\sx{2}{ \rot{ 22}$u\!=\!3$\ero }}
\put(051,06){\sx{2}{ \rot{ 53}$u\!=\!4$\ero }}

\put(082,183){\sx{2}{ \rot{2}$v\!=\!1$\ero }}
\put(081,083){\sx{2}{ \rot{46}$v\!=\!2$\ero }}
\put(306,148){\sx{2}{ \rot{0.}$v\!=\!0$\ero }}
\put(406,148){\sx{2}{ \rot{0.}$v\!=\!0$\ero }}
 \end{picture}
}
\end{document}

Warning

While only one Superfactorial is described, terms «Super Factorial», «SuperFactorial», «Superfactorial», «SuFac»
are treated as synonyms denoting the Superfunction of Factorial
constructed with Regular iteration at the fixed point \(L\!=\!2\)
that grows infinitely along the real axis and
has value 3 (minimal integer bigger than this \(L\)) at zero.

Factorial has many fixed points, and two of them are positive.

As soon as another Superfactorial be described, the notations need to be adjusted. Suggestion:

SuFac for that shown above, and AuFac for its inverse,
SdFac for that at the fixed point \(L\!=\!1\), and AdFac for its inverse,
SuperFac\(_n\) for another, constructed at the \(n\)th fixed point \(L_n\) of Factorial.

Let the colleague(s) who describes other super factorials (perhaps without fast growth along the real axis)
chose(s) the appropriate names top denote them.

References

  1. ↑ http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1
    http://mizugadro.mydns.jp/PAPERS/2010superfae.pdf reprint, English version
    http://mizugadro.mydns.jp/PAPERS/2010superfar.pdf reprint, Russian version
    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)
  2. ↑ https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics - Lambert Academic Publishing, 2020/7/28
  3. ↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.

Keywords

«[[]]», «Factorial», «Iterate», «Regular iteration», «SuFac», «SuperFactorial», «Superfactorial», «Superfunction», «Superfunctions», «[[]]»,

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current22:53, 22 August 2025Thumbnail for version as of 22:53, 22 August 20251,843 × 750 (835 KB)T (talk | contribs)Improve resolution, extend the field covered, add one contour line - as it is in the Book
17:50, 20 June 2013Thumbnail for version as of 17:50, 20 June 20131,142 × 453 (302 KB)Maintenance script (talk | contribs)Importing image file

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