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.

#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 "facp.cin"
#include "afacc.cin"
#include "abelfac.cin"
//#include "doya.cin"
//DB Shoko(DB x) { return log(1.+exp(x)*(M_E-1.)); }

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");
#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=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
}

Latex generator of labels

% %<br> % file IterFacPlot.pdf should be generated with the code above in order to compile the Latex document below. % % Copyleft 2012 by Dmitrii Kouznetsov <br> % \documentclass[12pt]{article} % <br> \usepackage{geometry} % <br> \usepackage{graphicx} % <br> \usepackage{rotating} % <br> \paperwidth 1008pt % <br> \paperheight 1008pt % <br> \topmargin -94pt % <br> \oddsidemargin -81pt % <br> \textwidth 1100pt % <br> \textheight 1100pt % <br> \pagestyle {empty} % <br> \newcommand \sx {\scalebox} % <br> \newcommand \rot {\begin{rotate}} % <br> \newcommand \ero {\end{rotate}} % <br> \newcommand \ing {\includegraphics} % <br> \parindent 0pt% <br> \pagestyle{empty} % <br> \begin{document} % <br> \begin{picture}(1002,1002) % <br> \put(10,10){\ing{IterFacPlot}} % <br> \put(11,978){\sx{3.4}{$y\!=\!\mathrm{Factorial}^c(x)$}} % <br> \put(11,898){\sx{4}{$9$}} % <br> \put(11,798){\sx{4}{$8$}} % <br> \put(11,698){\sx{4}{$7$}} % <br> \put(11,598){\sx{4}{$6$}} % <br> \put(11,498){\sx{4}{$5$}} % <br> \put(11,398){\sx{4}{$4$}} % <br> \put(11,298){\sx{4}{$3$}} % <br> \put(11,198){\sx{4}{$2$}} % <br> \put(11,098){\sx{4}{$1$}} % <br> % <br> \put(100,16){\sx{4}{$1$}} % <br> \put(200,16){\sx{4}{$2$}} % <br> \put(301,16){\sx{4}{$3$}} % <br> \put(401,16){\sx{4}{$4$}} % <br> \put(502,16){\sx{4}{$5$}} % <br> \put(602,16){\sx{4}{$6$}} % <br> \put(703,16){\sx{4}{$7$}} % <br> \put(803,16){\sx{4}{$8$}} % <br> \put(903,16){\sx{4}{$9$}} % <br> \put(990,16){\sx{4}{$x$}} % <br> % <br> \put(304,770){\sx{3.3}{\rot{87}$c\!=\!1.2$\ero}} % <br> \put(355,921){\sx{3.3}{\rot{84}$c\!=\!1$\ero}} % <br> \put(490,904){\sx{3.3}{\rot{74}$c\!=\!0.5$\ero}} % <br> \put(536,906){\sx{3.3}{\rot{70}$c\!=\!0.4$\ero}} % <br> \put(601,909){\sx{3.3}{\rot{66}$c\!=\!0.3$\ero}} % <br> \put(683,910){\sx{3.3}{\rot{63}$c\!=\!0.2$\ero}} % <br> \put(795,915){\sx{3.3}{\rot{55}$c\!=\!0.1$\ero}} % <br> % <br> \put(928,932){\sx{3.4}{\rot{45}$c\!=\!0$\ero}} % <br> % <br> \put(896,755){\sx{3.3}{\rot{35}$c\!=\!-0.1$\ero}} % <br> \put(892,646){\sx{3.3}{\rot{26}$c\!=\!-0.2$\ero}} % <br> \put(888,571){\sx{3.3}{\rot{19}$c\!=\!-0.3$\ero}} % <br> \put(884,513){\sx{3.3}{\rot{14}$c\!=\!-0.4$\ero}} % <br> \put(882,468){\sx{3.3}{\rot{11}$c\!=\!-0.5$\ero}} % <br> \put(882,430){\sx{3.3}{\rot{10}$c\!=\!-0.6$\ero}} % <br> % <br> \put(890,331){\sx{3.3}{\rot{3}$c\!=\!-1$\ero}} % <br> \put(876,290){\sx{3.3}{\rot{2}$c\!=\!-1.2$\ero}} % <br> \end{picture} % <br> \end{document} % <br> %