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.

This map is used as Fig.11.4 at page 138 of book «Superfunctions», 2020 ^[1][2]
in order to show the inverse function of the Transfer function considered in Chapter 11 as an example for the exotic iteration.

Generators

C++ generator of curves

Files ado.cin and conto.cin should be loaded to the working directory in order to compile 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);
 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(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,".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 %<br>
 \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{productlogmap}} %<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(4,118){ \sx{.8}{\rot{-36}$v\!=\!1.8$\ero}} %<br>
 \put(3,96){ \sx{.8}{\rot{-20}$v\!=\!2$\ero}} %<br>
 \put(2, 84){\sx{.6}{$0$}} %<br>
 \put(8, 84){\sx{.8}{\bf cut}} %<br>
 \put(7,72){\sx{.7}{\rot{18}$v\!=\!-\!2$\ero}} %<br>
 \put(-3,64){\sx{.6}{$-2$}} %<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( 41, 76){\rot{-39}\sx{.8}{$u\!=\!0.8$}\ero}%<br>
 \put( 41, 57){\rot{-26}\sx{.84}{$u\!=\!1$}\ero}%<br>
 \put( 39, 40){\rot{-19}\sx{.8}{$u\!=\!1.2$}\ero}%<br>
 \put( 33, 21){\rot{-18}\sx{.8}{$u\!=\!1.4$}\ero}%<br>
 \put( 85, 145){\rot{83}\sx{.82}{$v\!=\!1$}\ero}%<br>
 \put(137, 102){\rot{17}\sx{.8}{$v\!=\!0.2$}\ero}%<br>
 \put(144,  84){\rot{0}\sx{.8}{$v\!=\!0$}\ero}%<br>
 \put(140,  65){\rot{-16}\sx{.72}{$v\!=\!-\!0.2$}\ero}%<br>
 \put(134,  46){\rot{-32}\sx{.72}{$v\!=\!-0.4$}\ero}%<br>
 \put( 79,  33){\rot{-80}\sx{.86}{$v\!=\!-1$}\ero}%<br>
 \end{picture} %<br>
 } %<br>
 \end{document}

References

Keywords

«ArcLambertW», «ArcZex», «Exotic iteration», «Exotic iteration», «Fixed point», «LambertW», «ProductLog», «Superfunctions», «Transfer function», «Zex»,

«Суперфункции»,