File:DoyaPlotT100.png

Explicit plot of the Doya function

$ \mathrm{Doya}_t(z)=\mathrm{Tania}(t+\mathrm{ArcTania}(z))$

where the Tania function is solution of equations

$\displaystyle \mathrm{Tania}'(z)= \frac{ \mathrm{Tania}(z)}{1\!+\!\mathrm{Tania}(z)}~$, $~\mathrm{Tania}(0)\!=\!1$

At the picture, $y=\mathrm{Doya}_t(x)$ is plotted versus $x$ for various values of $t$.

Generators
The curves are generated with the C++ code. The resulting file is imported into the Latex document for generation of labels.

C++ generator of curves
using namespace std; typedef 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.)

z_type ArcTania(z_type z) {return z + log(z) - 1. ;}

z_type ArcTaniap(z_type z) {return 1. + 1./z ;}

z_type TaniaTay(z_type z) { int n; z_type s; s=1.+z*(.5+z*(1./16.+z*(-1./192.+z*(-1./3072.+z*(1.3/6144.+z*(-4.7/147456. //+z*(7.3/4128768.) //some reserve term )))))); DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }

z_type TaniaNega(z_type z){int n;z_type s=exp(z-exp(z)+1.); DO(n,4) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }

z_type TaniaBig(z_type z){int n;z_type s=z; s=z-log(s)+1.; DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; } z_type TaniaS(z_type z){int n; z_type s,t=z+z_type(2.,-M_PI);t*=2./9.; t=I*sqrt(t); s=-1.+t*(3.+t*(-3.+t*(.75+t*(.3+t*(.9/16.+t*(-.3/7.+t*(-12.51/224. //+t*(-.9/28.) ))))))); DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }

z_type Tania(z_type z){ z_type t; if( fabs(Im(z))< M_PI && Re(z)<-2.51) return TaniaNega(z); if( abs(z)>7. || Re(z)>3.8 ) return TaniaBig(z); if( Im(z) > .7 ) return TaniaS(z); if( Im(z) < -.7) return conj(TaniaS(conj(z))); return TaniaTay(z); }

void ado(FILE *O, int X, int Y) {     fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%'); fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y); fprintf(O,"/M {moveto} bind def\n"); fprintf(O,"/L {lineto} bind def\n"); fprintf(O,"/S {stroke} bind def\n"); fprintf(O,"/s {show newpath} bind def\n"); fprintf(O,"/C {closepath} bind def\n"); fprintf(O,"/F {fill} bind def\n"); fprintf(O,"/o {.1 0 360 arc C S} bind def\n"); fprintf(O,"/times-Roman findfont 20 scalefont setfont\n"); fprintf(O,"/W {setlinewidth} bind def\n"); fprintf(O,"/RGB {setrgbcolor} bind def\n");}


 * 1) define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);

main{ int j,k,m,n; DB x,y, a; FILE *o;o=fopen("doyaplot.eps","w");ado(o,408,408); fprintf(o,"4 4 translate\n 100 100 scale\n"); for(m=0;m<5;m++){ M(m,0)L(m,4)} for(n=0;n<5;n++){ M(0,n)L(4,n)} M(0,0)L(4,4) fprintf(o,".01 W 0 0 0 RGB S\n"); DO(n,181){x=.005+.01*n;y=Re(Tania(3.+ArcTania(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 0 .5 RGB S\n"); DO(n,249){x=.005+.01*n;y=Re(Tania(2.+ArcTania(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 0 .5 RGB S\n"); DO(n,163){x=.005+.02*n;y=Re(Tania(1.+ArcTania(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 0 .5 RGB S\n"); DO(n,101){x=.005+.04*n;y=Re(Tania(-1.+ArcTania(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W .5 0 0 RGB S\n"); DO(n,101){x=.005+.04*n;y=Re(Tania(-2.+ArcTania(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W .5 0 0 RGB S\n"); DO(n,101){x=.005+.04*n;y=Re(Tania(-3.+ArcTania(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W .5 0 0 RGB S\n"); fprintf(o,"showpage\n%cTrailer",'%'); fclose(o); system("epstopdf doyaplot.eps"); system(   "open doyaplot.pdf"); //these 2 commands may be specific for macintosh getchar; system("killall Preview");// if run at another operational sysetm, may need to modify }

Latex generator of the picture
\documentclass[12pt]{article} \usepackage{geometry} \usepackage{graphicx} \usepackage{rotating} % \paperwidth 419pt % \paperheight 426pt % \topmargin -103pt % \oddsidemargin -83pt % \textwidth 1200pt % \textheight 600pt % \pagestyle {empty} % \newcommand \sx {\scalebox} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \ing {\includegraphics} % \begin{document} % \sx{1}{ \begin{picture}(810,410) % \put(1,9){\ing{doyaplot}} % \put(-12,394){\sx{2.8}{$y$}} % \put(-12,303){\sx{2.8}{$3$}} % \put(-12,203){\sx{2.8}{$2$}} % \put(-12,103){\sx{2.8}{$1$}} % \put(0,-9){\sx{2.5}{$0$}} % \put(100,-9){\sx{2.5}{$1$}} % \put(200,-9){\sx{2.5}{$2$}} % \put(300,-9){\sx{2.5}{$3$}} % \put(392,-7){\sx{2.6}{$x$}} % %\put(560,214){\rot{37}\sx{4}{$y=\mathrm{Tania}(x)$}\ero} % \put(134,353){\rot{53}\sx{2.8}{$t\!=\!3$}\ero} % \put(194,354){\rot{50}\sx{2.8}{$t\!=\!2$}\ero} % \put(260,350){\rot{48}\sx{2.8}{$t\!=\!1$}\ero} % \put(336,350){\rot{45}\sx{2.8}{$t\!=\!0$}\ero} % \put(340,238){\rot{44}\sx{2.8}{$t\!=\!-1$}\ero} % \put(344,178){\rot{41}\sx{2.7}{$t\!=\!-2$}\ero} % \put(341,114){\rot{37}\sx{2.7}{$t\!=\!-3$}\ero} % \end{picture} % } % \end{document}

Copyleft status
This picture and its generators can be used for free; attribute the source.