File:IterfacPlotT.png

Explicit plot of iteration of Factorial, $y=\mathrm{Factorial}^c(x)$ versus $x$ for various values of number $c$ of iterations. The $c$th iteration of Factorial is implemented through the SuperFactorial and the AbelFactorial:
 * $\mathrm{Factorial}^c(x)=\mathrm{SuperFactorial}\Big(c+\mathrm{AbelFactorial}(x)\Big)$

The thick lines correspond to

$y=\mathrm{Factorial}(x)$, for $x\!>\!2$ it is realized at $c\!=\!1~$, and

$y=\mathrm{ArcFactorial}(x)$, for $x\!>\!2$ it is realized at $c\!=\!-1~$.

Warning:

$~y\!=\!\mathrm{Factorial}^c(x)~$ should not be confused with $~y\!=\!\mathrm{Factorial}(x^c)~$, nor with $~y\!=\!\mathrm{Factorial}(x)^c~$.

C++ generator of curves
// Files ado.cin, fac.cin, superfactorial.cin, facp.cin, afacc.cin, abelfac.cin should be loaded.

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"
 * 7) include "facp.cin"
 * 8) include "afacc.cin"
 * 9) include "abelfac.cin"

main{ int m,n; double x,y,t; FILE *o; o=fopen("IterFacPlot.eps","w"); ado(o,1010,1010); 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"); // 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<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"); for(m=0;m<42;m++){x=0.+.1*m; y=Re(fac(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 1 0 RGB S\n"); for(m=0;m<30;m++){x=0.8856031944+(1./90.)*m*m; y=Re(afacc(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 1 0 1 RGB S\n"); for(n=-12;n<13;n++){t=.1*n; M(2,2); DO(m,802){x=2.01+.01*m; y=Re(abelfac(x)); y=Re(superfac(t+y)); L(x,y); if(y>10.1)break;} } fprintf(o,"1 setlinecap 1 setlinejoin .02 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf IterFacPlot.eps"); system(   "open IterFacPlot.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 IterFacPlot.pdf should be generated with the code above in order to compile the Latex document below. % % Copyleft 2012 by Dmitrii Kouznetsov % \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{IterFacPlot}} % \put(11,978){\sx{3.4}{$y\!=\!\mathrm{Factorial}^c(x)$}} % \put(11,898){\sx{4}{$9$}} % \put(11,798){\sx{4}{$8$}} % \put(11,698){\sx{4}{$7$}} % \put(11,598){\sx{4}{$6$}} % \put(11,498){\sx{4}{$5$}} % \put(11,398){\sx{4}{$4$}} % \put(11,298){\sx{4}{$3$}} % \put(11,198){\sx{4}{$2$}} % \put(11,098){\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(304,770){\sx{3.3}{\rot{87}$c\!=\!1.2$\ero}} % \put(355,921){\sx{3.3}{\rot{84}$c\!=\!1$\ero}} % \put(490,904){\sx{3.3}{\rot{74}$c\!=\!0.5$\ero}} % \put(536,906){\sx{3.3}{\rot{70}$c\!=\!0.4$\ero}} % \put(601,909){\sx{3.3}{\rot{66}$c\!=\!0.3$\ero}} % \put(683,910){\sx{3.3}{\rot{63}$c\!=\!0.2$\ero}} % \put(795,915){\sx{3.3}{\rot{55}$c\!=\!0.1$\ero}} % % \put(928,932){\sx{3.4}{\rot{45}$c\!=\!0$\ero}} % % \put(896,755){\sx{3.3}{\rot{35}$c\!=\!-0.1$\ero}} % \put(892,646){\sx{3.3}{\rot{26}$c\!=\!-0.2$\ero}} % \put(888,571){\sx{3.3}{\rot{19}$c\!=\!-0.3$\ero}} % \put(884,513){\sx{3.3}{\rot{14}$c\!=\!-0.4$\ero}} % \put(882,468){\sx{3.3}{\rot{11}$c\!=\!-0.5$\ero}} % \put(882,430){\sx{3.3}{\rot{10}$c\!=\!-0.6$\ero}} % % \put(890,331){\sx{3.3}{\rot{3}$c\!=\!-1$\ero}} % \put(876,290){\sx{3.3}{\rot{2}$c\!=\!-1.2$\ero}} % \end{picture} % \end{document} % %