Difference between revisions of "File:FacitT.jpg"

Iterates of Factorial:

\(y=\mathrm{Factorial}^n(x)\ \) versus \(x\) for various \(n\).

The iterates are expressed through the Superfunciton SuFac and the Abelfunction SuFac:

\[ \mathrm{Factorial}^n(z)=\mathrm{SuFac}\big(n+\mathrm{AuFac}(zbig) \]

These functions are described in the Moscow University Physics Bulletin, 2010 ^[1].

This plot is used as Fig.8.9 at page 100 of book «Superfunctions»^[2][3]
in order to show the smooth transition from Factorial through the Identity Function to the ArcFactorial.

C++ generator of curves

// Do not forget to load all the *.cin mentioned below in the end of the first paragraph of the code!

#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 "facp.cin"
#include "afacc.cin"
#include "superfac.cin"
#include "abelfac.cin"
#include "ado.cin"

int main(){ int m,n; DB x,y, p,q, t; z_type z,c,d;
#define M(x,y) fprintf(o,"%5.3f %5.3f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%5.3f %5.3f L\n",0.+x,0.+y);
FILE *o;o=fopen("facit.eps","w");ado(o,1002,1002);
fprintf(o,"1 1  translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");

DO(m,11){M(m,0)L(m,10)}
DO(n,11){M(0,n)L(10, n)}
fprintf(o,".01 W S\n");

DO(n,2001){x=.004*n; z=x; c=fac(z); y=Re(c); if(y>10.06) break; if(n==0)M(x,y) else L(x,y);} fprintf(o,".05 W 0 .9 0 RGB S\n");
DO(n,2001){x=.002*n; z=x; c=fac(fac(z)); y=Re(c); if(y>10.06) break; if(n==0)M(x,y) else L(x,y);} fprintf(o,".05 W 0 .9 0 RGB S\n");
DO(n,2001){x=.002*n; z=x; c=fac(fac(fac(z))); y=Re(c); if(y>10.06) break; if(n==0)M(x,y) else L(x,y);} fprintf(o,".05 W 0 .9 0 RGB S\n");
M(0,1)L(2,1)L(2,10)fprintf(o,".03 W 0 .8 0 RGB S\n");
DO(n,1001){x=.88+.01*n;z=x; c=afacc(z); y=Re(c); if(x>10.02||y>10.06) break; if(n==0)M(x,y) else L(x,y);} fprintf(o,".05 W .9 0 .9 RGB S\n");
DO(n,1001){x=.95+.01*n;z=x; c=afacc(afacc(z)); y=Re(c); if(x>10.02||y>10.06) break; if(n==0)M(x,y) else L(x,y);} fprintf(o,".05 W .9 0 .9 RGB S\n");
DO(n,1001){x=.98+.01*n;z=x; c=afacc(afacc(afacc(z))); y=Re(c); if(x>10.02||y>10.06) break; if(n==0)M(x,y) else L(x,y);} fprintf(o,".05 W .9 0 .9 RGB S\n");
M(1,.48)L(1,2)L(10,2) fprintf(o,".03 W .9 0 .9 RGB S\n");

DO(m,41){ t=-2.+.1*m;
M(1,1);DO(n,1901){x=1.0005+.005*n; c=abelfac(x); c=superfac(t+c); y=Re(c);if(y>10.06) break; L(x,y);}fprintf(o,".012 W 0 0 0 RGB S\n");}

fprintf(o,"showpage\n\%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf facit.eps"); 
        system(    "open facit.pdf"); // macintosh
        getchar(); system("killall Preview"); // macintosh
}

Latex generator of labels

\documentclass[12pt]{article}
\paperwidth 1008pt
\paperheight 1008pt
\textwidth 1800pt
\textheight 1800pt
\topmargin -108pt
\oddsidemargin -72pt 
\parindent 0pt
\pagestyle{empty}
\usepackage {graphics}
\usepackage{rotating}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}
\begin{document}
\begin{picture}(1006,1006) \put(0,0){\ing{facit}}
\put(4,984){\sx{4}{$y$}}
\put(4,890){\sx{4}{$9$}}
\put(4,790){\sx{4}{$8$}}
\put(4,690){\sx{4}{$7$}}
\put(4,590){\sx{4}{$6$}}
\put(4,490){\sx{4}{$5$}}
\put(4,390){\sx{4}{$4$}}
\put(4,290){\sx{4}{$3$}}
\put(4,190){\sx{4}{$2$}}
\put(4, 90){\sx{4}{$1$}}
\put(92,4){\sx{4}{$1$}}
\put(192,4){\sx{4}{$2$}}
\put(292,4){\sx{4}{$3$}}
\put(392,4){\sx{4}{$4$}}
\put(492,4){\sx{4}{$5$}}
\put(592,4){\sx{4}{$6$}}
\put(692,4){\sx{4}{$7$}}
\put(792,4){\sx{4}{$8$}}
\put(892,4){\sx{4}{$9$}}
\put(982,4){\sx{4.1}{$x$}}
\put(212,830){\rot{89}\sx{5}{$n\!\rightarrow\!\infty$}\ero}
\put(262,824){\rot{87}\sx{5}{$n\!=\!2$}\ero}
\put(340,816){\rot{83}\sx{5}{$n\!=\!1$}\ero}
\put(454,806){\rot{78}\sx{5}{$n\!=\!0.5$}\ero}
\put(494,806){\rot{74}\sx{5}{$n\!=\!0.4$}\ero}
\put(544,806){\rot{68}\sx{5}{$n\!=\!0.3$}\ero}
\put(606,806){\rot{61}\sx{5}{$n\!=\!0.2$}\ero}
\put(698,806){\rot{54}\sx{5}{$n\!=\!0.1$}\ero}
\put(766,744){\rot{45}\sx{5.2}{$n\!=\!0$}\ero}
\put(770,638){\rot{34}\sx{5}{$n\!=\!-0.1$}\ero}
\put(770,556){\rot{27}\sx{5}{$n\!=\!-0.2$}\ero}
\put(770,494){\rot{21}\sx{5}{$n\!=\!-0.3$}\ero}
\put(770,450){\rot{15}\sx{5}{$n\!=\!-0.4$}\ero}
\put(770,414){\rot{11}\sx{5}{$n\!=\!-0.5$}\ero}
\put(770,311){\rot{4}\sx{5}{$n\!=\!-1$}\ero}
\put(770,232){\rot{1}\sx{5}{$n\!=\!-2$}\ero}
\put(770,183){\rot{.01}\sx{5}{$n\!\rightarrow\!-\infty$}\ero}
\end{picture}
\end{document}

References

Keywords

«Abelfactorial», «Abelfunction», «AuFac», «Factorial», «Iterate», «Regular iteration», «SuFac», «Superfactorial», «Superfunction», «Superfunctions», «Square root of factorial»,

«Корень из факториала», «Суперфункции»,