File:LambertWmap150.png

Complex map of the LambertW function.

$f=\mathrm{LambertW}(x\!+\!\mathrm i y)$ is shown with levels

$\Re(f)=u=\mathrm{const~}$ and $\Im(f)=v=\mathrm{const~}$.

The representation through the Tania function is used.

C++ generator of curves
Files ado.cin and conto.cin should be loaded to the working directory in order to compile the code below:

using namespace std; typedef complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) 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); 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-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=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 .5 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 .5 1 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf productlogmap.eps"); system(   "open productlogmap.pdf"); getchar; system("killall Preview"); }

Latex generator of lables
% 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{productlogmap}} % \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}