File:Facite.jpg

Iterates of Factorial,

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

For non–integer value sof $n$, the iterate is expressed through functions SuFac and AuFac:

$\mathrm{Factorial}^n(z)=\mathrm{SuFac}\Big(n+\mathrm{AuFac}(z)\Big)$

C++ generator of curves
// do not forget to load ado.cin, fac.cin, facp.cin, afacc.cin, SuFac.cin, AuFac.cin

//using namespace std; typedef std::complex z_type; //DB B=2.; //DB F(DB z) { return exp( exp( log(B)*z));} //DB G(DB z) { return log( log(z) )/log(B);} int main{ int m,n; double x,y,t; FILE *o; o=fopen("FacIte.eps","w"); ado(o,1020,1020); fprintf(o,"1 1 translate 100 100 scale\n"); for(m=0;m<11;m++) {M(m,0)L(m,10)} for(m=0;m<11;m++) {M(0,m)L(10,m)} fprintf(o,"2 setlinecap .01 W S\n"); DO(m,402){x=0.001+.01*m; y=Re(fac(x)); if(m==0)M(x,y) else L(x,y); if(y>10.1) break;} DO(m,402){x=0.001+.01*m; y=Re(fac(fac(x))); if(m==0)M(x,y) else L(x,y); if(y>10.1) break;} DO(m,402){x=0.001+.01*m; y=Re(fac(fac(fac(x)))); if(m==0)M(x,y) else L(x,y); if(y>10.1) break;} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 1 1 RGB S\n"); DO(m,1002){x=.88+.01*m; y=Re(afacc(x)); if(m==0)M(x,y) else L(x,y);} DO(m,1002){x=.95+.01*m; y=Re(afacc(afacc(x))); if(m==0)M(x,y) else L(x,y);} DO(m,1002){x=.98+.01*m; y=Re(afacc(afacc(afacc(x)))); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .03 W 1 .5 0 RGB S\n"); M(0,0)L(10,10) fprintf(o,"1 setlinecap 1 setlinejoin .03 W 1 0 1 RGB S\n");
 * 1) include
 * 2) include
 * 3) include
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x10.1||y>10.1)break;} } fprintf(o,"1 setlinecap 1 setlinejoin .002 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf facite.eps"); system(   "open facite.pdf"); getchar; system("killall Preview"); }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \usepackage{graphicx} \usepackage{rotating} \paperwidth 1008pt \paperheight 1008pt \topmargin -94pt \oddsidemargin -81pt \textwidth 1100pt \textheight 1100pt \pagestyle {empty} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \parindent 0pt% \pagestyle{empty} \begin{document} \begin{picture}(1002,1002) %\put(10,10){\ing{IterPowPlot}} \put(10,10){\ing{FacIte}} %\put(11,976){\sx{4}{$y\!=\!\mathrm{pow}_2^c(x)$}} \put(13,991){\sx{4.4}{$y$}}% \!=\!\mathrm{Factorial}^n(x)$}} \put(13,898){\sx{4}{$9$}} \put(13,798){\sx{4}{$8$}} \put(13,698){\sx{4}{$7$}} \put(13,598){\sx{4}{$6$}} \put(13,498){\sx{4}{$5$}} \put(13,398){\sx{4}{$4$}} \put(13,298){\sx{4}{$3$}} \put(13,198){\sx{4}{$2$}} \put(13,098){\sx{4}{$1$}} \put(100,16){\sx{4}{$1$}} \put(200,16){\sx{4}{$2$}} \put(301,16){\sx{4}{$3$}} \put(401,16){\sx{4}{$4$}} \put(502,16){\sx{4}{$5$}} \put(602,16){\sx{4}{$6$}} \put(703,16){\sx{4}{$7$}} \put(803,16){\sx{4}{$8$}} \put(903,16){\sx{4}{$9$}} \put(990,16){\sx{4}{$x$}} \put(230,920){\sx{3.6}{\rot{89}$n\!=\!3$\ero}} \put(272,920){\sx{3.6}{\rot{88}$n\!=\!2$\ero}} \put(355,921){\sx{3.6}{\rot{85}$n\!=\!1$\ero}} \put(342,791){\sx{2.9}{\rot{84}$y\!=\!x!$\ero}} \put(441,890){\sx{3.6}{\rot{79}$n\!=\!0.6$\ero}} \put(473,890){\sx{3.6}{\rot{74}$n\!=\!0.5$\ero}} \put(530,894){\sx{3.6}{\rot{70}$n\!=\!0.4$\ero}} \put(598,894){\sx{3.6}{\rot{69}$n\!=\!0.3$\ero}} \put(674,898){\sx{3.6}{\rot{62}$n\!=\!0.2$\ero}} \put(792,910){\sx{3.6}{\rot{54}$n\!=\!0.1$\ero}} \put(928,932){\sx{3.6}{\rot{45}$n\!=\!0$\ero}} \put(738,722){\sx{3.6}{\rot{45}$y\!=\!x$\ero}} % \put(888,744){\sx{3.6}{\rot{36}$n\!=\!-0.1$\ero}} \put(877,634){\sx{3.6}{\rot{28}$n\!=\!-0.2$\ero}} \put(871,560){\sx{3.6}{\rot{20}$n\!=\!-0.3$\ero}} \put(866,504){\sx{3.6}{\rot{16}$n\!=\!-0.4$\ero}} \put(866,456){\sx{3.6}{\rot{14}$n\!=\!-0.5$\ero}} \put(862,411){\sx{3.6}{\rot{10}$n\!=\!-0.6$\ero}} \put(889,328){\sx{3.6}{\rot{5}$n\!=\!-1$\ero}} \put(886,250){\sx{3.6}{\rot{0}$n\!=\!-2$\ero}} \put(886,212){\sx{3.6}{\rot{0}$n\!=\!-3$\ero}} \end{picture} \end{document}