File:TaniaNegMapT.png

File
File history
File usage
Metadata

Size of this preview: 607 × 600 pixels. Other resolutions: 243 × 240 pixels | 1,773 × 1,752 pixels.

Original file ‎(1,773 × 1,752 pixels, file size: 306 KB, MIME type: image/png)

Complex map of the truncation of the asymptotic expansion of the Tania function for the domain between the cut lines.

For $|\Im(z)| < \pi$, $\Re(z)\rightarrow - \infty$,

$ \mathrm{Tania}(z) ~ \sim ~

\varepsilon

-\varepsilon^2+\frac{3}{2}\varepsilon^3 -\frac{8}{3}\varepsilon^4+\frac{125}{24}\varepsilon^6+ O(\varepsilon^6) $ where $\varepsilon=\exp(1+z)$

The truncation of the series gives the approximation shown in the figure.

The map shows

$ f=\varepsilon

-\varepsilon^2+\frac{3}{2}\varepsilon^3 -\frac{7}{2}\varepsilon^4 $

in the plane $x=\Re(z)$, $y=\Im(z)$; the lines $u=\Re(f)=\mathrm{const}$ and the lines $v=\Im(f)=\mathrm{const}$ are drawn.

The shading indicate the region with precision worse than 3 decimal digits

Generators

Common header

Files conto.cin and ado.cin should be in 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 TaniaNeg(z_type z){int n; z_type e=exp(1.+z);
return e*(1.+e*(-1.+e*(1.5+e*(-8./3.+e*(125./24. ) )))); }
 
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);
}

C++ generator of the shading

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("tanianegmapD2.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=TaniaNeg(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=16;q=.5;
conto(o,g,w,v,X,Y,M,N, (3),-p,p);
L(-8.,8) L(8,8) L(8,-8) L(-8,-8) 
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 tanianegmapD2.eps");   
       system(    "open tanianegmapD2.pdf");
       getchar(); system("killall Preview");
}

C++ generator of the 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("tanianegmap1.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=TaniaNeg(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 tanianegmap1.eps");    
       system(    "open tanianegmap1.pdf");
       getchar(); system("killall Preview");
}

Latex generator of the lables

% Gerenator of TaniaNegMap.png % 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{taniacontour}} % \put(6,5){\ing{taniaNegmapD2}} % \put(6,5){\ing{taniaNegmap1}} % \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(20, 84){\sx{.8}{\bf cut}} % \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{86}\sx{.8}{$u\!=\!1$}\ero}% % \put(106, 77){\rot{87}\sx{.8}{$u\!=\!2$}\ero}% % \put(121, 77){\rot{88}\sx{.8}{$u\!=\!3$}\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}

File history

Click on a date/time to view the file as it appeared at that time.

	Date/Time	Thumbnail	Dimensions	User	Comment
current	17:50, 20 June 2013		1,773 × 1,752 (306 KB)	Maintenance script (talk \| contribs)	Importing image file

You cannot overwrite this file.

File usage

The following page links to this file:

Tania function

Metadata

This file contains additional information, probably added from the digital camera or scanner used to create or digitize it. If the file has been modified from its original state, some details may not fully reflect the modified file.

Horizontal resolution	59.06 dpc
Vertical resolution	59.06 dpc

File:TaniaNegMapT.png

Contents

Generators

Common header

C++ generator of the shading

C++ generator of the curves

Latex generator of the lables

File history

File usage

Metadata

Navigation menu

Personal tools

Namespaces

Variants

Views

More

Search

.

Tools