File:AfacplotT2px300.png

Explicit plot of ArcFactorial

$y=\mathrm{ArcFctorial}(x)$

(solid thick line) and plot of its asymptotic

$ \displaystyle y= \mathrm{Bart}+\mathrm{Liza}_1\,(x\!-\!\mathrm{Homer})^{1/2}+\mathrm{Liza}_2\,(x\!-\!\mathrm{Homer})$

(thin black line).

$\mathrm{Bart} ~ ~\approx~ 0.4616321449683622$

$\mathrm{Homer}\! \approx\! 0.8856031944108887$

$\mathrm{Liza}_1 ~ \approx 1.5276760433847776$

$\mathrm{Liza}_2 ~ \approx 0.3559463008501492$

C++ generator of curves

// Files fac.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"

main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
FILE *o;o=fopen("afacplot.eps","w");ado(o,620,310);
#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 1 translate\n 100 100 scale\n");
fprintf(o,"2 setlinejoin 2 setlinecap\n");
DO(m,7){M(m,0)L(m,3)}
DO(n,4){M(0,n)L(6,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,73) { x=Homer+.001*(m*m+.5); y=Re(afacc(x)); L(x,y);}
fprintf(o,"0 0 1 RGB .03 W S\n");
M(Homer,Bart)
DO(m,39) { x=Homer+.001*(m*m+.5); y=Bart+
  Liza1*sqrt(x-Homer)+
  Liza2*(x-Homer);
  L(x,y);}
M(Homer,Bart)
DO(m,62) { x=Homer+.001*(m*m+.5); y=Bart+
  Liza1*sqrt(x-Homer)+
  Liza2*(x-Homer)+
  Liza3*(x-Homer)*sqrt(x-Homer) ;
  L(x,y);}
fprintf(o,"0 0 0 RGB .006 W S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
       system("epstopdf afacplot.eps");        
       system(    "open afacplot.pdf");        //for LINUX
//     getchar(); system("killall Preview");//for mac
}

Latex generator of labels

% %<br> \documentclass[12pt]{article} %<br> \usepackage{geometry} %<br> \usepackage{graphicx} %<br> \usepackage{rotating} %<br> \usepackage{hyperref} %<br> \paperwidth 1216px %<br> \paperheight 608px %<br> \textwidth 1666mm %<br> \textheight 1333mm %<br> \topmargin -107pt %<br> \oddsidemargin -72pt %<br> \parindent 0pt %<br> \begin {document} %<br> \newcommand \sx {\scalebox} %<br> \newcommand \rme {{e}} %<br> \newcommand \rmi {{\rm i}} %imaginary unity is always roman font %<br> \newcommand \ds {\displaystyle} %<br> \newcommand \rot {\begin{rotate}} %<br> \newcommand \ero {\end{rotate}} %<br> \sx{2}{\begin{picture}(640,303) %<br> \put(0,0){\includegraphics{afacplot}} %<br> \put( 3,290){\sx{2}{$y$}} %<br> \put( 3,194){\sx{2}{\bf 2}} %<br> \put( 3, 94){\sx{2}{\bf 1}} %<br> \put( 3, 43){\sx{1.2}{Bart}} %<br> %\put( 93, 2){\sx{1.2}{\rot{90}Homer\ero}} %<br> \put( 54, 4){\sx{1.2}{Homer}} %<br> \put( 96, 3){\sx{2}{\bf 1}} %<br> \put(196, 3){\sx{2}{\bf 2}} %<br> \put(296, 3){\sx{2}{\bf 3}} %<br> \put(396, 3){\sx{2}{\bf 4}} %<br> \put(496, 3){\sx{2}{\bf 5}} %<br> \put(588, 3){\sx{2.2}{$x$}} %<br> \put(111,286){\sx{1.2}{$y\!=\! \mathrm{Bart}+\mathrm{Liza}_1 (x\!-\!\mathrm{Homer})^{1/2}+\mathrm{Liza}_2 (x\!-\!\mathrm{Homer})$}} \put(276,226){\sx{1.3}{$y\!=\! \mathrm{ArcFactorial}(x)$}} \put(188,133){\sx{1.1}{$y\!=\! \mathrm{Bart}+\mathrm{Liza}_1 (x\!-\!\mathrm{Homer})^{1/2}+\mathrm{Liza}_2 (x\!-\!\mathrm{Homer})+\mathrm{Liza}_3 (x\!-\!\mathrm{Homer})^{3/2}$}} %\put(158,348){\rot{-73}\sx{1.5}{$v\!=\!1.2$}\ero} %<br> \end{picture}} %<br> \end{document} %<br> %