File:Aufact.png

Explicit plot of function AuFac (blue), compared to ArcFactorial (red)

C++ generator of curves
// Files fac.cin, facp.cin, afacc.cin, ado.cin should be loaded in order to compile the code below

// // using namespace std; typedef std::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 "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"

int 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); 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");
 * 1) define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);

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
\documentclass[12pt]{article} \paperwidth 808pt \paperheight 638pt \textwidth 1800pt \textheight 1800pt \topmargin -72pt \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}(806,580) \begin{picture}(806,606) %\put(0,0){\includegraphics{SuperFacPlot}} % \put(0,0){\includegraphics{AbelFacPlo}} \put(9,593){\sx{4}{$y$}} \put(9,501){\sx{4}{$2$}} \put(9,401){\sx{4}{$1$}} \put(9,301){\sx{4}{$0$}} \put(-2,201){\sx{4}{$-\!1$}} \put(-2,101){\sx{4}{$-\!2$}} \put(093,282){\sx{4}{$1$}} \put(193,282){\sx{4}{$2$}} \put(293,282){\sx{4}{$3$}} \put(393,282){\sx{4}{$4$}} \put(493,282){\sx{4}{$5$}} \put(593,282){\sx{4}{$6$}} \put(693,282){\sx{4}{$7$}} \put(782,282){\sx{4}{$x$}} %\put(352,540){\sx{4.2}{\rot{8.8}$y\!=\!\mathrm{ArcFactorial}(x)$\ero}} \put(324,524){\sx{5}{\rot{8.8}$y\!=\!\mathrm{ArcFactorial}(x)$\ero}} %\put(660,450){\sx{4}{\rot{83}$y\!=\!\mathrm{Factorial}(x)$\ero}} % %\put(350,380){\sx{4.2}{\rot{10}$y\!=\!\mathrm{AuFac}(x)$\ero}} \put(380,390){\sx{5}{\rot{10}$y\!=\!\mathrm{AuFac}(x)$\ero}} \end{picture} \end{document}