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:

using namespace std; typedef complex z_type; // #include "facp.cin" // #include "afacc.cin"
 * 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"
 * 1) 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); 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 }
 * 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);

Latex generator of labels
% % \documentclass[12pt]{article} % \usepackage{geometry} % \usepackage{graphicx} % \usepackage{rotating} % \usepackage{hyperref} % \paperwidth 1216px % \paperheight 608px % \textwidth 1666mm % \textheight 1333mm % \topmargin -107pt % \oddsidemargin -72pt % \parindent 0pt % \begin {document} % \newcommand \sx {\scalebox} % \newcommand \rme 	 % \newcommand \rmi 	%imaginary unity is always roman font % \newcommand \ds {\displaystyle} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \sx{2}{\begin{picture}(640,303) % \put(0,0){\includegraphics{afacplot}} % \put( 3,290){\sx{2}{$y$}} % \put( 3,194){\sx{2}{\bf 2}} % \put( 3, 94){\sx{2}{\bf 1}} % \put( 3, 43){\sx{1.2}{Bart}} % %\put( 93, 2){\sx{1.2}{\rot{90}Homer\ero}} % \put( 54, 4){\sx{1.2}{Homer}} % \put( 96, 3){\sx{2}{\bf 1}} % \put(196, 3){\sx{2}{\bf 2}} % \put(296, 3){\sx{2}{\bf 3}} % \put(396, 3){\sx{2}{\bf 4}} % \put(496, 3){\sx{2}{\bf 5}} % \put(588, 3){\sx{2.2}{$x$}} % \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} % \end{picture}} % \end{document} % %