File:DoyaconT70.png

Contour plot of the Doya function versus argument and parameter.

$f=\mathrm{Doya}_t(x)$ is shown in the $x,t$ plane with levels $f\!=\!\mathrm{const}$.

Generator of curves

The files ado.cin and conto.cin should be loaded to the working directory in order to compile the 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 "conto.cin"

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.) //reserve term for testing
)))))); DO(n,2) 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,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }

z_type TaniaBig0(z_type z){int n;z_type  L=log(z), s=z-L+1.; 
s-=(1.-L)/z;  
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 TaniaBig0(z);
if( Im(z) > .7 ) return TaniaS(z);
if( Im(z) < -.7) return conj(TaniaS(conj(z)));
return TaniaTay(z);
}

z_type Doya(z_type t, z_type z){ return Tania(t+ArcTania(z)) ;}

main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
 int M=80,M1=M+1;
 int N=80,N1=N+1;
DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
char v[M1*N1]; // v is working array
FILE *o;o=fopen("doyacon1.eps","w");ado(o,42,82);
fprintf(o,"1 41 translate\n 10 10 scale\n");
DO(m,M1) X[m]=0.005+.05*(m);
DO(n,N1) Y[n]=-4.+.1*n;
for(m= 0;m<5;m++){M(m,-4)L(m,4)}
for(n=-4;n<5;n++){M(  0,n)L(4,n)}
fprintf(o,".011 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
DO(m,M1){x=X[m]; //printf("%5.2f\n",x);
DO(n,N1){y=Y[n]; //z=z_type(x,y);	
 c=Doya(y,x); 
 p=Re(c);//q=Im(c);	
 if(p>-99. && p<99.  ){ g[m*N1+n]=p;}
        }}
fprintf(o,"1 setlinejoin 2 setlinecap\n");  p=3.;q=2.;
for(m=0;m<1;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".02 W 0 0 .9 RGB S\n");
for(m=-9;m<8;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 0 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
	system("epstopdf doyacon1.eps");	
	system(    "open doyacon1.pdf");
	getchar(); system("killall Preview");
}

Generator of labels

File doyacon1.pdf should be generated with the code from the previous section in order to pdflatex the document below.

% Copyleft 2011 by Dmitrii Kouznetsov <br> \documentclass[12pt]{article} %<br> \usepackage{geometry} %<br> \usepackage{graphicx} %<br> \usepackage{rotating} %<br> \paperwidth 456pt %<br> \paperheight 843pt %<br> \topmargin -94pt %<br> \oddsidemargin -96pt %<br> \textwidth 1200pt %<br> \textheight 1800pt %<br> \pagestyle {empty} %<br> \newcommand \sx {\scalebox} %<br> \newcommand \rot {\begin{rotate}} %<br> \newcommand \ero {\end{rotate}} %<br> \newcommand \ing {\includegraphics} %<br> \begin{document} %<br> \sx{10}{ \begin{picture}(41,81) %<br> \put(1,1){\ing{doyacon1}} % <br> \put(-.5,79.6){\sx{.4}{$t$}}% <br> \put(-1,71){\sx{.4}{$3$}}% <br> \put(-1,61){\sx{.4}{$2$}}% <br> \put(-1,51){\sx{.4}{$1$}}% <br> \put(-1,41){\sx{.4}{$0$}}% <br> \put(-4,31){\sx{.4}{$-1$}}% <br> \put(-4,21){\sx{.4}{$-2$}}% <br> \put(-4,11){\sx{.4}{$-3$}}% <br> %\put(-4, 1){\sx{.4}{$-4$}}% <br> \put( 1,-1.5){\sx{.4}{0}}% <br> \put(11,-1.5){\sx{.4}{1}}% <br> \put(21,-1.5){\sx{.4}{2}}% <br> \put(31,-1.5){\sx{.4}{3}}% <br> \put(39.8,-1.2){\sx{.42}{$x$}}% <br> \put(9.7,16){\rot{-62}\sx{.33}{$f\!=\!0.2$}\ero}% <br> \put(14.,18){\rot{-60}\sx{.33}{$f\!=\!0.4$}\ero}% <br> \put(25.4,22){\rot{-56}\sx{.33}{$f\!=\!1$}\ero}% <br> \put(30.4,25.5){\rot{-55}\sx{.33}{$f\!=\!2$}\ero} % <br> \put(30.4,40){\rot{-55}\sx{.33}{$f\!=\!3$}\ero} % <br> \put(31.3,51){\rot{-54}\sx{.33}{$f\!=\!4$}\ero} % <br> \put(31.6,62){\rot{-53}\sx{.33}{$f\!=\!5$}\ero} % <br> \put(32.8,73){\rot{-52}\sx{.33}{$f\!=\!6$}\ero} % <br> \put(35,81.6){\rot{-51}\sx{.33}{$f\!=\!7$}\ero} % <br> \end{picture} %<br> } %<br> \end{document}