File:FilogmT1000a.jpg

Complex map of function Filog.

$u\!+\!\mathrm i v= \mathrm{Filog} (x\!+\!\mathrm i y)$

Meaning
$z=\mathrm{Filog}(a)~$ expresses solution $z$ of equation $\ln(z)/a=z$.

C++ generator of curves
using namespace std; typedef complex z_type; int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; int M=600,M1=M+1; int N=401,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("filogzoo.eps","w");ado(o,304,204); fprintf(o,"102 102 translate\n 100 100 scale\n"); DO(m,M1) X[m]=-1.+.005*(m-.2); DO(n,200)Y[n]=-1.+.005*n; Y[200]=-.0001; Y[201]= .0001; for(n=202;n-15. && p<15. && q>-15. && q<15. ){ g[m*N1+n]=p;f[m*N1+n]=q;} }} fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=4.;q=1; 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,".001 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,".001 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,".001 W 0 0 .9 RGB S\n"); for(m=1;m<14;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".004 W .9 0 0 RGB S\n"); for(m=1;m<14;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".004 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".004 W .6 0 .6 RGB S\n"); for(m=-11;m<14;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".004 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf filogzoo.eps"); system(   "open filogzoo.pdf"); //for mac //   getchar; system("killall Preview"); // for mac // Copyleft 2012 by Dmitrii Kouznetsov }
 * 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"
 * 5) include "filog.cin"

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \paperwidth 315pt \paperheight 216pt \topmargin -102pt \oddsidemargin -88pt \usepackage{graphicx} \usepackage{rotating} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \rme {\mathrm{e}} \newcommand \sx {\scalebox} \begin{document} \begin{picture}(208,208) \put(8,8){\includegraphics{filogzoo}} \put(4,207){$1$} \put(0,147){$y$} \put(4,107){$0$} \put(-3, 8){$-\!1$} \put(2, 0){$-\!1$} \put(108, 0){$0$} \put(140, 2){\sx{.8}{$1/\rme$}} \put(209, 0){\sx{1}{$1$}} \put(264, 2){\sx{.8}{$\pi/2$}} \put(307, 2){\sx{.9}{$x$}} \put(185,43){\sx{.8}{\rot{64}$v\!=\!2$ \ero} } \put(250,49){\sx{.6}{\rot{84}$v\!=\!1.2$ \ero} } \put(281.5,43){\sx{.8}{\rot{80}$v\!=\!1$ \ero} } \put(283,161){\sx{.6}{\rot{-40}$v\!=\!0.8$ \ero} } \put(250,143){\sx{.6}{\rot{60}$u\!=\!0.2$ \ero} } \put(283,118){\sx{.8}{\rot{35}$u\!=\!0$ \ero} } \put(284,74){\sx{.6}{\rot{5}$u\!=\!-0.2$ \ero} } \put(28.4,120){\sx{.7}{\rot{79}$u\!=\!0.6$\ero}} \put(84.4,113){\sx{.7}{\rot{74}$u\!=\!0.8$\ero}} \end{picture} \end{document}