File:Figlogzo2t.jpg

Complex map of function Filog, zoomin of the square of size unity from the central part of figure http://mizugadro.mydns.jp/t/index.php/File:Filogbigmap100.png

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

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

C++ generator of curves
// Files ado.cin, conto.cin, filog.cin, tania.cin should be loaded to the working directory in order to compile the C++ code below 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=500,M1=M+1; int N=501,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("filogzo2.eps","w");ado(o,1020,1020); fprintf(o,"510 510 translate\n 1000 1000 scale\n"); DO(m,M1) X[m]=-.5+.002*(m-.2); DO(n,250)Y[n]=-.5+.002*n; Y[250]=-.00001; Y[251]= .00001; for(n=252;n
 * 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"

for(m=-5;m<6;m++){M(.1*m,-.5)L(.1*m,.5)} for(n=-5;n<6;n++){M( -.5,.1*n)L(.5,.1*n)} fprintf(o,"2 setlinecap .001 W 0 0 0 RGB S\n"); M(exp(-1.),-.5) L(exp(-1.), .5) //M(M_PI/2.,-.5) //L(M_PI/2., .5) fprintf(o,".0007 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(z_type(-1.,-M_PI)+log(z))/(-z); c=Filog(z); p=Re(c);q=Im(c); if(p>-201. && p<201. && q>-201. && q<201. ){ g[m*N1+n]=p;f[m*N1+n]=q;} }} fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=5.;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,".0008 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,".0008 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,".0008 W 0 0 .9 RGB S\n"); for(m=1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".002 W .9 0 0 RGB S\n"); for(m=1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".002 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".002 W .6 0 .6 RGB S\n"); for(m=-20;m<21;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".002 W 0 0 0 RGB S\n"); p=10.; conto(o,g,w,v,X,Y,M,N, 30 ,-p,p); fprintf(o,".002 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, 40 ,-p,p); fprintf(o,".002 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, 50 ,-p,p); fprintf(o,".002 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,-30 ,-p,p); fprintf(o,".002 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,-40 ,-p,p); fprintf(o,".002 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,-50 ,-p,p); fprintf(o,".002 W 0 0 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, -30,-p,p); fprintf(o,".002 W .9 0 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, -40,-p,p); fprintf(o,".002 W .9 0 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, -50,-p,p); fprintf(o,".002 W .9 0 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, 30,-p,p); fprintf(o,".002 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, 40,-p,p); fprintf(o,".002 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, 50,-p,p); fprintf(o,".002 W 0 0 .9 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf filogzo2.eps"); system(   "open filogzo2.pdf"); //for mac //   getchar; system("killall Preview"); // for mac // Copyleft 2012 by Dmitrii Kouznetsov }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \paperwidth 1030pt \paperheight 1030pt \textwidth 1140pt \textheight 1140pt \topmargin -103pt \oddsidemargin -86pt \usepackage{graphicx} \usepackage{rotating} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \rme {\mathrm{e}} \newcommand \sx {\scalebox} \begin{document} \begin{picture}(1020,1020) \put(10,10){\includegraphics{filogzo2}}

\put(0,1012){\sx{2.3}{$y$}} \put(-7,914){\sx{2.3}{$\frac{4}{10}$}} \put(-7,814){\sx{2.3}{$\frac{3}{10}$}} \put(-7,714){\sx{2.3}{$\frac{2}{10}$}} \put(-7,614){\sx{2.3}{$\frac{1}{10}$}} \put( 2,514){\sx{2.3}{$0$}} \put(-7,414){\sx{2.3}{$\frac{-\!1}{10}$}} \put(-7,314){\sx{2.3}{$\frac{-\!1}{5}$}} \put(-7,214){\sx{2.3}{$\frac{-\!3}{10}$}} \put(-7,114){\sx{2.3}{$\frac{-\!2}{5}$}} \put(-7, 14){\sx{2.3}{$\frac{-\!1}{2}$}} \put(90, -1){\sx{2.3}{$-0.4$}} \put(190, -1){\sx{2.3}{$-0.3$}} \put(290, -1){\sx{2.3}{$-0.2$}} \put(390, -1){\sx{2.3}{$-0.1$}} \put(516, -1){\sx{2.3}{$0$}} \put(609, -1){\sx{2.3}{$0.1$}} \put(709, -1){\sx{2.3}{$0.2$}} \put(809, -1){\sx{2.3}{$0.3$}} \put(866, -1){\sx{2.2}{$1/\mathrm e$}} \put(910, -1){\sx{2.3}{$0.4$}} \put(1010, 0){\sx{2.4}{$x$}} \put(250,540){\sx{3}{\rot{82}$u\!=\!0.8$ \ero} } \put(530,540){\sx{3}{\rot{81}$u\!=\!1$ \ero} } \put(680,524){\sx{2.8}{\rot{77}$u\!=\!1.2$ \ero} } \put(768,524){\sx{2.8}{\rot{72}$u\!=\!1.4$ \ero} } \put(816,524){\sx{2.8}{\rot{56}$u\!=\!1.6$ \ero} } \put(600,684){\sx{2.8}{\rot{-33}$v\!=\!0.2$ \ero} } \put(720,798){\sx{2.8}{\rot{-62}$v\!=\!0.4$ \ero} } \put(820,490){\sx{3}{\rot{-4}$v\!=\!1$ \ero} } \put(820,466){\sx{3}{\rot{-12}$v\!=\!2$ \ero} } \put(808,446){\sx{3}{\rot{-26}$v\!=\!3$ \ero} } \put(768,372){\sx{3}{\rot{36}$u\!=\!5$ \ero} } \put(770,350){\sx{3}{\rot{26}$u\!=\!4$ \ero} } \put(770,324){\sx{3}{\rot{16}$u\!=\!3$ \ero} } \put(768,297){\sx{3}{\rot{2}$u\!=\!2$ \ero} } \put(766,262){\sx{3}{\rot{-8}$u\!=\!1$ \ero} } \put(752,230){\sx{3}{\rot{-28}$u\!=\!0$ \ero} }

\put(94,99){\sx{3}{\rot{41}$v\!=\!-2$ \ero} } \put(162,50){\sx{3}{\rot{57}$v\!=\!-1$ \ero} } \put(242,30){\sx{3}{\rot{75}$v\!=\!0$ \ero} } \put(315,30){\sx{3}{\rot{87}$v\!=\!1$ \ero} }

\end{picture} \end{document}