File:TaniaBigMap.png

The complex map of the truncated expansion of the Tania function at large values of its argument. Function
$f=\mathrm{Tania}(z)=$ $ (z\!+\!1)\!-\!\ln(z\!+\!1) +\frac{ \ln(z\!+\!1)}{z+1}$ $ +\big(\frac{ \ln(z\!+\!1)}{z+1}\big)^{\!2} \big(\frac{1}{2}-\ln(z\!+\!1)^{-1}\big) +\big(\frac{ \ln(z\!+\!1)}{z+1}\big)^{\!3} \big(

                   \frac{1}{3}-\frac{3}{4} \ln(z\!+\!1)^{-1}+ \ln(z\!+\!1)^{-2}\big)$ $

+\big(\frac{ \ln(z\!+\!1)}{z+1}\big)^{\!4} \big(

                   \frac{1}{4}-\frac{11}{\!5} \ln(z\!+\!1)^{-1}+3\ln(z\!+\!1)^{-2}-\ln(z\!+\!1)^{-3}\big)$ 

is shown in coordinates $z=x+\mathrm i y$ with lines $u=\Re(f)=\mathrm {const}$ and lines $v=\Im(f)=\mathrm {const}$.

In the shaded range, the precision of the approximation is smaller than 3; the precision is defined with
$\mathrm{Precision}(z)=$ $ \displaystyle - \lg\big( \frac{ |\mathrm{Tania}(z)-\mathrm{approximation}(z)|} { |\mathrm{Tania}(z)|+|\mathrm{approximation}(z)|} \big) $
and indicates, how many significant digits does the truncated series return.

The approximation fails at small values of $z$, and also for $z$ between the cut lines of the Tania function, id est, x<0, $|y| \le \pi$,

Generators

Common header with routines

Files conto.cin, ado.cin should be loaded to the working directory.

#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 t=1.+z;
z_type L=log(t); 
z_type x=L/t;
z_type m=1./L;
z_type s = t-L + x*(1. + x*( .5-m + x*( 1./3. + m*(-1.5+m) +x*( .25 +m*(-11./6.+m*(3.-m)) 
//      +x*(.2 +m*(-25./12 +m*(35./6. +m*(-5. +m)))) //reserve term for the testing
))));
//DO(n,2) 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;  return s ;
DO(n,4) s+=(z-ArcTania(s))/ArcTaniap(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);
}

Generator of curves

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("taniabigmap.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]=-.03;
        Y[81]= .03;
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=TaniaBig(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 RGB S\n");
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 RGB S\n");
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 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 .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 taniabigmap.eps");     
       system(    "open taniabigmap.pdf");
       getchar(); system("killall Preview");
}

Generator of the shaded region

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=160,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("taniabigmapD2.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,N1)Y[n]=-8.+.1*(n);
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=TaniaBig(z);
 d=Tania(z);
//  c=ArcTania(c);
 p=-log(  abs(c-d) / (abs(c)+abs(d)) )/log(10.) ;
 //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;
conto(o,g,w,v,X,Y,M,N, (3),-p,p); fprintf(o," 1 .9 .9 RGB C F\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 taniabigmapD2.eps");   
       system(    "open taniabigmapD2.pdf");
       getchar(); system("killall Preview");
}

Generator of the labels

FIles taniabigmapD2.pdf and taniabigmap.pdf should be already generated with the codes from the sections above.

\documentclass[12pt]{article} %<br> \usepackage{geometry} %<br> \usepackage{graphicx} %<br> \usepackage{rotating} %<br> \paperwidth 854pt %<br> \paperheight 844pt %<br> \topmargin -96pt %<br> \oddsidemargin -98pt %<br> \textwidth 1100pt %<br> \textheight 1100pt %<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{5}{ \begin{picture}(164,165) %<br> % \put(6,5){\ing{taniacontour}} %<br> \put(6,5){\ing{taniabigmapD2}} %<br> \put(6,5){\ing{taniabigmap}} %<br> \put(2,162){\sx{.7}{$y$}} %<br> \put(2,144){\sx{.6}{$6$}} %<br> \put(2,124){\sx{.6}{$4$}} %<br> \put(2,104){\sx{.6}{$2$}} %<br> %\put(3,116){\sx{.6}{$\pi$ ~ \bf cut}} %<br> %\put(23,100){\sx{.8}{$u\!=\!0$}} %<br> \put(2, 84){\sx{.6}{$0$}} %<br> % \put(20, 84){\sx{.8}{$v\!=\!0$}} %<br> \put(20, 84){\sx{.8}{\bf cut}} %<br> %\put(23,68){\sx{.8}{$u\!=\!0$}} %<br> \put(-3,64){\sx{.6}{$-2$}} %<br> %\put(-3,53){\sx{.6}{$-\pi$ ~ \bf cut}} %<br> \put(-3,44){\sx{.6}{$-4$}} %<br> \put(-3,24){\sx{.6}{$-6$}} %<br> \put( 22,0){\sx{.6}{$-6$}} %<br> \put( 42,0){\sx{.6}{$-4$}} %<br> \put( 62,0){\sx{.6}{$-2$}} %<br> \put( 86,0){\sx{.6}{$0$}} %<br> \put(106,0){\sx{.6}{$2$}} %<br> \put(126,0){\sx{.6}{$4$}} %<br> \put(146,0){\sx{.6}{$6$}} %<br> \put(164,0){\sx{.7}{$x$}} %<br> \put(139,154){\rot{-6}\sx{.8}{$v\!=\!6$}\ero}%<br> \put(138,143){\rot{-6}\sx{.8}{$v\!=\!5$}\ero}%<br> \put(137,132){\rot{-6}\sx{.8}{$v\!=\!4$}\ero}%<br> \put(136,121){\rot{-6}\sx{.8}{$v\!=\!3$}\ero}%<br> \put(135,109){\rot{-5}\sx{.8}{$v\!=\!2$}\ero}%<br> \put( 89, 83){\rot{86}\sx{.8}{$u\!=\!1$}\ero}%<br> \put(106, 77){\rot{87}\sx{.8}{$u\!=\!2$}\ero}%<br> \put(121, 77){\rot{88}\sx{.8}{$u\!=\!3$}\ero}%<br> \put(134, 97){\rot{-4}\sx{.8}{$v\!=\!1$}\ero}%<br> \put(134, 84){\rot{0}\sx{.8}{$v\!=\!0$}\ero}%<br> \put(134, 72){\rot{3}\sx{.72}{$v\!=\!-\!1$}\ero}%<br> \put(133, 60){\rot{3}\sx{.72}{$v\!=\!-\!2$}\ero}%<br> \put(134, 48){\rot{3}\sx{.72}{$v\!=\!-\!3$}\ero}%<br> \put(135, 36){\rot{3}\sx{.72}{$v\!=\!-\!4$}\ero}%<br> \put(136, 25){\rot{3}\sx{.72}{$v\!=\!-\!5$}\ero}%<br> \put(137, 14){\rot{3}\sx{.72}{$v\!=\!-\!6$}\ero}%<br> \end{picture} %<br> } %<br> \end{document}

Copyright status

Copyleft 2011 by Dmitrii Kouznetsov. You may use the image and its generators for free, but attribute the source.