File:AbelFacPloT.png

Explicit plot of ArcFactorial and AbelFactorial.

$\mathrm{ArcFactorial}=\mathrm{Factorial}^{-1}$

$\mathrm{AbelFactorial}=\mathrm{SuperFactorial}^{-1}$

C++ generator of curves

Files fac.cin, facp.cin, afacc.cin and ado.cin should be loaded in the working directory in order to compile the C++ code below.

#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 "fac.cin"
#include "facp.cin"
#include "afacc.cin"
#include "ado.cin"

z_type arcsuperfac0(z_type z){ int n; z_type s, c, e;
DB k=0.61278745233070836381366079016859252;
DB U[19]={1.,                          -0.798731835172434541585621072345730147,
0.69806411355936704552792746483306691, -0.6339640557572814865638000833478131,  
0.5884152357911398848274232132172143,  -0.5538887519936519511632593654732843,  
0.526547902598592454703287733600892,   -0.504191460428021561516069870422848,   
0.48545298002933922263549078734881,    -0.46943468090947139273094056497701,
0.4555204862393622788179080677150,     -0.4432726222110411295132308010077,     
0.4323708863150174727399798603985,     -0.4225752531177612936293974175008,     
0.413701949171132722406449918702,      -0.40560764595293667778491699902, 
0.39817872478532299454624349817,       -0.391323,      0.384};
//     z-=2.; s=U[15]*z; for(n=14;n>=0;n--){ s+=U[n]; s*=z;}
       z-=2.; s=U[18]*z; for(n=17;n>=0;n--){ s+=U[n]; s*=z;}
return log(s)/k;}

z_type arcsuperfac(z_type z){ if(abs(z-2.)<.12) return arcsuperfac0(z);
                               return  arcsuperfac(afacc(z))+1.;} // As in the Paper

z_type abelfac(z_type z){ if(abs(z-2.)<.12) return arcsuperfac0(z)+0.91938596545217788;
                               return  abelfac(afacc(z))+1.;}

// #include "arcsuperfac.cin"

main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
FILE *o;o=fopen("AbelFacPlo.eps","w");ado(o,810,640);
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
fprintf(o,"1 315 translate\n 100 100 scale\n");
fprintf(o,"2 setlinejoin 2 setlinecap\n");
DO(m,9){M(m,-3)L(m,3)} for(n=-3;n<4;n++){M(0,n)L(8,n)}   fprintf(o,".006 W 0 0 0 RGB S\n");

DB Bart=0.4616321449683622;
DB Homer=0.8856031944108887;
DB Liza1=1.5276760433847776;
DB Liza2=0.3559463008501492;
DB Liza3=-0.4620189870305121;

M(0,Bart)L(Homer,Bart)L(Homer,0)  fprintf(o,".004 W 0 0 0 RGB S\n");
fprintf(o,"1 setlinejoin 1 setlinecap\n");

M(Homer,Bart)
DO(m,86) { x=Homer+.001*(m*m+.5); y=Re(afacc(x)); L(x,y);}  fprintf(o,"1 0 0 RGB .03 W S\n");
/* M(Homer,Bart)
DO(m,38) { x=Homer+.001*(m*m+.5); y=Bart+
  Liza1*sqrt(x-Homer)+
  Liza2*(x-Homer);
  L(x,y);}
fprintf(o,"0 0 0 RGB .006 W S\n"); // Expansion of ArcFactorial at the branch point 
*/
DO(m,61){t=m/60.; x=2.09+6.*t*t; y=Re(abelfac(x)); if(m==0)M(x,y) else L(x,y);}
fprintf(o,"0 0 1 RGB .03 W S\n"); 
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
       system("epstopdf AbelFacPlo.eps");      
       system(    "open AbelFacPlo.pdf");      //for LINUX
//     getchar(); system("killall Preview");//for mac
}

Latex generator of labels

% File AbelFacPlo.pdf should be generated with the code above in order to compile the Latex docuent below

% %<br> \documentclass[12pt]{article} %<br> \usepackage{geometry} %<br> \usepackage{graphics} %<br> \usepackage{rotating} %<br> \paperwidth 806pt %<br> \paperheight 630pt %<br> \topmargin -96pt %<br> \oddsidemargin -72pt %<br> \textwidth 1204pt %<br> \textheight 1204pt %<br> \newcommand \sx {\scalebox} %<br> \newcommand \ing \includegraphics %<br> \newcommand \rot {\begin{rotate}} %<br> \newcommand \ero {\end{rotate}} %<br> \parindent 0pt %<br> \pagestyle{empty} %<br> \begin{document} %<br> \begin{picture}(806,630)%<br> %\put(0,0){\includegraphics{SuperFacPlot}} %<br> \put(0,0){\includegraphics{AbelFacPlo}} %<br> \put(9,626){\sx{3}{$y$}} %<br> \put(9,505){\sx{3}{$2$}} %<br> \put(9,405){\sx{3}{$1$}} %<br> \put(9,305){\sx{3}{$0$}} %<br> \put(-2,205){\sx{3}{$-\!1$}} %<br> \put(-2,105){\sx{3}{$-\!2$}} %<br> \put(094,290){\sx{3}{$1$}} %<br> \put(194,290){\sx{3}{$2$}} %<br> \put(294,290){\sx{3}{$3$}} %<br> \put(394,290){\sx{3}{$4$}} %<br> \put(494,290){\sx{3}{$5$}} %<br> \put(594,290){\sx{3}{$6$}} %<br> \put(694,290){\sx{3}{$7$}} %<br> \put(784,291){\sx{3}{$x$}} %<br> \put(352,540){\sx{4.2}{\rot{8.8}$y\!=\!\mathrm{ArcFactorial}(x)$\ero}} %<br> %\put(660,450){\sx{4}{\rot{83}$y\!=\!\mathrm{Factorial}(x)$\ero}} %<br> \put(350,380){\sx{4.2}{\rot{10}$y\!=\!\mathrm{AbelFactorial}(x)$\ero}} %<br> \end{picture} %<br> \end{document} %<br> %