File:TaniaMap100.png

Summary

Title / Description	Complex map of the Tania function, ${\displaystyle y=\mathrm {Tania} (x)}$.
Citizendium author & Copyright holder	Copyright © Dmitrii Kouznetsov. See below for licence/re-use information.
Date created	2011 September 25
Country of first publication	Japan
Notes	The generator of the figure uses files conto.cin and ado.cin
Other versions	http://tori.ils.uec.ac.jp/TORI/index.php/File:TaniaContourPlot100.png
Using this image on CZ	Please click here to add the credit line, then copy the code below to add this image to a Citizendium article, changing the size, alignment, and caption as necessary. {{Image|TaniaMap100.png|right|350px|Add image caption here.}}

Licensing

   

This media, TaniaMap100.png, is licenced under the Creative Commons Attribution 3.0 Unported License

You are free: To Share — To copy, distribute and transmit the work; To Remix — To adapt the work.
Under the following conditions: Attribution — You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).
For any reuse or distribution, you must make clear to others the licence terms of this work (the best way to do this is with a link to this licence's web page). Any of the above conditions can be waived if you get permission from the copyright holder. Nothing in this licence impairs or restricts the author's moral rights.
Read the full licence.

Generation

First, the curves are generated with C++ code below.

Then, the latex document uses these curves to generate the figure.

Then, the result can be converted to some *.png or any other format.

C++ generator of curves

Files ado.cin and conto.cin are necessary for compilation of the generator of the curves 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.) //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);
}

main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=160,M1=M+1;
int N=161,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("taniamap.eps","w");ado(o,162,162);
fprintf(o,"81 81 translate\n 10 10 scale\n");
DO(m,M1) X[m]=-8.+.1*(m);
DO(n,80)Y[n]=-8.+.1*n;
       Y[80]=-.033;
       Y[81]= .033;
for(n=82;n<N1;n++) Y[n]=-8.+.1*(n-1.);
for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}}
for(n=-8;n<9;n++){     M(  -8,n)L(8,n)}
fprintf(o,".008 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=Tania(z); p=Re(c);q=Im(c);
if(p>-99. && p<99. &&  q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 2 setlinecap\n");  p=.6;q=.5;
for(m=-10;m<10;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".01 W 0 .6 0 $
for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".01 W .9 0 0 $
for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".01 W 0 0 .9 $
for(m=1;m<10;m++)  conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".05 W .9 0 0 RGB S\n");
for(m=1;m<10;m++)  conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 .9 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".05 W .6 0 .6 RGB S\n");
for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".05 W 0 0 0 RGB S\n");
y= M_PI; for(m=0;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)}
y=-M_PI; for(m=0;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)}
fprintf(o,".07 W 1 .5 0 RGB S\n");
y= M_PI; for(m=2;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)}
y=-M_PI; for(m=2;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)}
fprintf(o,".07 W 0 .5 1 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
      system("epstopdf taniamap.eps");
      system(    "open taniamap.pdf");
      getchar(); system("killall Preview");
}

Latex generator of the labels

% Copyleft 2011 by Dmitrii Kouznetsov

\documentclass[12pt]{article} 
\usepackage{geometry} 
\usepackage{graphicx} 
\usepackage{rotating} 
\paperwidth 854pt 
\paperheight 844pt 
\topmargin -96pt 
\oddsidemargin -98pt 
\textwidth 1100pt 
\textheight 1100pt 
\pagestyle {empty} 
\newcommand \sx {\scalebox} 
\newcommand \rot {\begin{rotate}} 
\newcommand \ero {\end{rotate}} 
\newcommand \ing {\includegraphics} 
\begin{document} 
\sx{5}{ \begin{picture}(164,165) 
\put(6,5){\ing{taniamap}} 
\put(2,162){\sx{.7}{$y$}} 
\put(2,144){\sx{.6}{$6$}} 
\put(2,124){\sx{.6}{$4$}} 
\put(2,104){\sx{.6}{$2$}} 
\put(3,116){\sx{.6}{$\pi$ ~ \bf cut}} 
\put(23,100){\sx{.8}{$u\!=\!0$}} 
\put(2, 84){\sx{.6}{$0$}} 
\put(20, 84){\sx{.8}{$v\!=\!0$}} 
\put(23,68){\sx{.8}{$u\!=\!0$}} 
\put(-3,64){\sx{.6}{$-2$}} 
\put(-3,53){\sx{.6}{$-\pi$ ~ \bf cut}} 
\put(-3,44){\sx{.6}{$-4$}} 
\put(-3,24){\sx{.6}{$-6$}} 
\put( 22,0){\sx{.6}{$-6$}} 
\put( 42,0){\sx{.6}{$-4$}} 
\put( 62,0){\sx{.6}{$-2$}} 
\put( 86,0){\sx{.6}{$0$}} 
\put(106,0){\sx{.6}{$2$}} 
\put(126,0){\sx{.6}{$4$}} 
\put(146,0){\sx{.6}{$6$}} 
\put(164,0){\sx{.7}{$x$}} 
\put(139,154){\rot{-6}\sx{.8}{$v\!=\!6$}\ero}
\put(138,143){\rot{-6}\sx{.8}{$v\!=\!5$}\ero}
\put(137,132){\rot{-6}\sx{.8}{$v\!=\!4$}\ero}
\put(136,121){\rot{-6}\sx{.8}{$v\!=\!3$}\ero}
\put(135,109){\rot{-5}\sx{.8}{$v\!=\!2$}\ero}
\put( 89, 83){\rot{82}\sx{.8}{$u\!=\!1$}\ero}
\put(106, 81){\rot{84}\sx{.8}{$u\!=\!2$}\ero}
\put(134, 97){\rot{-4}\sx{.8}{$v\!=\!1$}\ero}
\put(134, 84){\rot{0}\sx{.8}{$v\!=\!0$}\ero}
\put(134, 72){\rot{3}\sx{.72}{$v\!=\!-\!1$}\ero}
\put(133, 60){\rot{3}\sx{.72}{$v\!=\!-\!2$}\ero}
\put(134, 48){\rot{3}\sx{.72}{$v\!=\!-\!3$}\ero}
\put(135, 36){\rot{3}\sx{.72}{$v\!=\!-\!4$}\ero}
\put(136, 25){\rot{3}\sx{.72}{$v\!=\!-\!5$}\ero}
\put(137, 14){\rot{3}\sx{.72}{$v\!=\!-\!6$}\ero}
\end{picture} 
} 
\end{document}