File:SuperFacPlotT.png

SuperFactorial of real argument (Blue curve) in comparison with Factorial (Red curve).

Factorial is holomorphic solution of equations

$\mathrm{Factorial}(z\!+\!1)=(z\!+\!1)\, \mathrm{Factorial}(z)~$, $~\mathrm{Factorial}(0)\!=\!1$

SuperFactorial$(z)=\mathrm{Factorial}^z(3)$ is holomorphic solution of equations

$\mathrm{SuperFactorial}(z\!+\!1)=\mathrm{Factorial}\Big(\mathrm{SuperFactorial}(z)\Big)~$ , $~\mathrm{SuperFactorial}(0)\!=\!3$

Generator of curves

// Files fac.cin, superfactorial.cin and ado.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 "ado.cin"
#include "fac.cin"
#include "superfactorial.cin"
//#include "doya.cin"
//DB Shoko(DB x) { return log(1.+exp(x)*(M_E-1.)); }

main(){ int m,n; double x,y; FILE *o;
o=fopen("SuperFacPlot.eps","w"); ado(o,802,1010);
fprintf(o,"401 1 translate 100 100 scale\n");
#define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);
for(m=-4;m<5;m++) {M(m,0)L(m,10)}
for(m=0;m<11;m++) {M(-4,m)L(4,m)}
fprintf(o,"2 setlinecap .01 W S\n");
// for(m=0;m<81;m++){x=-4.+.1*m; y=Shoko(x); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 0.6 0 RGB S\n");
// for(m=0;m<81;m++){x=-4.+.1*m; y=Re(Tania(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .014 W 0.4 0 .4 RGB S\n");
  for(m=0;m<42;m++){x=-.5+.1*m; y=Re(fac(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 1 0 0 RGB S\n");
  for(m=0;m<54;m++){x=-4+.1*m; y=Re(superfac(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 0 1 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
   system("epstopdf SuperFacPlot.eps");
   system(    "open SuperFacPlot.pdf");            
   getchar(); system("killall Preview");//for mac
}

Latex generator of labels

% File SuperFacPlot.pdf should be generated with the code above in order to compile the Latex document below:

% \documentclass[12pt]{article} %<br> \usepackage{geometry} %<br> \usepackage{graphics} %<br> \usepackage{rotating} %<br> \paperwidth 806pt %<br> \paperheight 1016pt %<br> \topmargin -96pt %<br> \oddsidemargin -72pt %<br> \textwidth 1004pt %<br> \textheight 1400pt %<br> \newcommand \sx {\scalebox} %<br> \newcommand \ing \includegraphics %<br> \newcommand \rot {\begin{rotate}} %<br> \newcommand \ero {\end{rotate}} %<br> \parindent 0pt %<br> \pagestyle{empty} %<br> \begin{document} %<br> \begin{picture}(602,1002) %<br> \put(0,0){\includegraphics{SuperFacPlot}} %<br> \put(380,999){\sx{3}{$y$}} %<br> \put(380,891){\sx{3}{$9$}} %<br> \put(380,791){\sx{3}{$8$}} %<br> \put(380,691){\sx{3}{$7$}} %<br> \put(380,591){\sx{3}{$6$}} %<br> \put(380,491){\sx{3}{$5$}} %<br> \put(380,391){\sx{3}{$4$}} %<br> \put(380,291){\sx{3}{$3$}} %<br> \put(380,191){\sx{3}{$2$}} %<br> \put(380,91){\sx{3}{$1$}} %<br> \put( 77,5){\sx{3}{$-\!3$}} %<br> \put(177,5){\sx{3}{$-\!2$}} %<br> \put(277,5){\sx{3}{$-\!1$}} %<br> \put(394,5){\sx{3}{$0$}} %<br> \put(494,5){\sx{3}{$1$}} %<br> \put(594,5){\sx{3}{$2$}} %<br> \put(694,5){\sx{3}{$3$}} %<br> \put(782,5){\sx{3}{$x$}} %<br> \put(532,510){\sx{4}{\rot{83}$y\!=\!\mathrm{SuperFactorial}(x)$\ero}} %<br> %\put(660,450){\sx{4}{\rot{83}$y\!=\!\mathrm{Factorial}(x)$\ero}} %<br> \put(740,550){\sx{4}{\rot{83}$y\!=\!\mathrm{Factorial}(x)$\ero}} %<br> \end{picture} %<br> \end{document} %<br> %