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.

using namespace std; typedef complex z_type; //#include "doya.cin" //DB Shoko(DB x) { return log(1.+exp(x)*(M_E-1.)); }
 * 1) include
 * 2) include
 * 3) include
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "ado.cin"
 * 5) include "fac.cin"
 * 6) include "superfactorial.cin"

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"); 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 }
 * 1) define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);

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