File:LambertWmapT.png

Summary

Licensing

   

This media, LambertWmapT.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.
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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.

Generators

First, the curves are generated with the C++ program. The principal branch of the Lambert W function is easy to evaluate through the Tania function, the code is supplied below. Then, the labels are added in the Latex document. The result is converted to PNG or any other appropriate format.

C++ generator of the curves

Files conto.cin and ado.cin should be in the working directory for the compilation of 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.) //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("productlogmap.eps","w");ado(o,162,162); 
FILE *o;o=fopen("lambertwmap.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=Tania(log(z)-1.); 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,".03 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,".03 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,".03 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,".07 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,".07 W 0 0 .9 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".07 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,".07 W 0 0 0 RGB S\n");
y=0.; for(m=0;m<75;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)}
fprintf(o,".07 W 1 .6 0 RGB S\n");
y=0.; for(m=2;m<75;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)}
fprintf(o,".07 W 0 .6 1 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
      system("epstopdf lambertwmap.eps");   
      system(    "open lambertwmap.pdf");
      getchar(); system("killall Preview");
}

Latex document generating the labels

\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{productlogmap}} %
\put(6,5){\ing{lambertwmap}} %
\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(4,118){ \sx{.8}{\rot{-36}$v\!=\!1.8$\ero}} %
\put(3,96){ \sx{.8}{\rot{-20}$v\!=\!2$\ero}} %
\put(2, 84){\sx{.6}{$0$}} %
\put(8, 84){\sx{.8}{\bf cut}} %
\put(7,72){\sx{.7}{\rot{18}$v\!=\!-\!2$\ero}} %
\put(-3,64){\sx{.6}{$-2$}} %
\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( 41, 76){\rot{-39}\sx{.8}{$u\!=\!0.8$}\ero}%
\put( 41, 57){\rot{-26}\sx{.84}{$u\!=\!1$}\ero}%
\put( 39, 40){\rot{-19}\sx{.8}{$u\!=\!1.2$}\ero}%
\put( 33, 21){\rot{-18}\sx{.8}{$u\!=\!1.4$}\ero}%
\put( 85, 145){\rot{83}\sx{.82}{$v\!=\!1$}\ero}%
\put(137, 102){\rot{17}\sx{.8}{$v\!=\!0.2$}\ero}%
\put(144, 84){\rot{0}\sx{.8}{$v\!=\!0$}\ero}%
\put(140, 65){\rot{-16}\sx{.72}{$v\!=\!-\!0.2$}\ero}%
\put(134, 46){\rot{-32}\sx{.72}{$v\!=\!-0.4$}\ero}%
\put( 79, 33){\rot{-80}\sx{.86}{$v\!=\!-1$}\ero}%
\end{picture} %
} %
\end{document}

% Copyleft 2011 by Dmitrii Kouznetsov. The free use is allowed, but, please, attribute the source. It is not for any "priority", but to trace errors if any. Dmitrii Kouznetsov 02:08, 26 September 2011 (UTC)