File:FacitT.jpg

Iterates of Factorial:

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

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

using namespace std; typedef complex z_type;
 * 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 "fac.cin"
 * 5) include "facp.cin"
 * 6) include "afacc.cin"
 * 7) include "superfac.cin"
 * 8) include "abelfac.cin"
 * 9) include "ado.cin"

int main{ int m,n; DB x,y, p,q, t; z_type z,c,d; 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");
 * 1) define M(x,y) fprintf(o,"%5.3f %5.3f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%5.3f %5.3f L\n",0.+x,0.+y);

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}