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Complex map of the ArcTania function,

$ \mathrm{ArcTania}(z)= z+\ln(z) -1$

$f=\mathrm{ArcTania}(x\!+\!\mathrm{i} y)$ in the $x,y$ plane; lines $u=\Re(f)=\mathrm{const}$ and lines $v=\Im(f)=\mathrm{const}$ are drawn. The integer values correspond to thick lines.


C++ generator of curves

Files ado.cin and conto.cin are necessary 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. ;}
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("arctaniacontour.eps","w");ado(o,162,162);
fprintf(o,"81 81 translate\n 10 10 scale\n");
DO(m,M1) X[m]=-8.+.1*(m);
       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,".0009 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);
//     p=abs(c-d)/(abs(c)+abs(d));  p=-log(p)/log(10.)-1.;
       if(p>-99. && p<99. &&
//       (fabs(y)>.034 ||x>-.9 ||fabs(x-int(x))>1.e-3) &&
          q>-99. && q<99 //&& fabs(q)> 1.e-19
fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1.8;q=.7;
for(m=-11;m<11;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".011 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,".011 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,".011 W 0 0 .9 RGB S\n");
for(m=1;m<11;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<11;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<80;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<80;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 arctaniacontour.eps");
       system(    "open arctaniacontour.pdf");            
       getchar(); system("killall Preview");//for mac

Latex Generator of lables

 % Gerenator of ArcTaniaMap.png %<br>
 % 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{arctaniacontour}} %<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(23,100){\sx{.8}{$u\!=\!0$}} %<br>
 \put(2, 84){\sx{.6}{$0$}} %<br>
 \put(59,85){\sx{.6}{\bf cut}} %<br>
% \put(20, 84){\sx{.8}{$v\!=\!0$}} %<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( 81, 23){\rot{81}\sx{.8}{$u\!=\!0$}\ero}%<br>
 \put( 92, 23){\rot{82}\sx{.8}{$u\!=\!1$}\ero}%<br>
 \put(101, 22){\rot{82}\sx{.8}{$u\!=\!2$}\ero}%<br>
 \put(111, 21){\rot{83}\sx{.8}{$u\!=\!3$}\ero}%<br>
 \put(120, 21){\rot{84}\sx{.8}{$u\!=\!4$}\ero}%<br>

 \put(134, 92){\rot{3}\sx{.8}{$v\!=\!1$}\ero}%<br>
 \put(134, 84){\rot{0}\sx{.8}{$v\!=\!0$}\ero}%<br>
 \put(134, 76){\rot{-3}\sx{.8}{$v\!=\!-\!1$}\ero}%<br>
 \put(133, 68){\rot{-5}\sx{.8}{$v\!=\!-\!2$}\ero}%<br>
 \put(134, 59){\rot{-5}\sx{.8}{$v\!=\!-\!3$}\ero}%<br>
 \put(135, 51){\rot{-5}\sx{.8}{$v\!=\!-\!4$}\ero}%<br>
 \put(135, 41){\rot{-5}\sx{.8}{$v\!=\!-\!5$}\ero}%<br>
 \put(135, 32){\rot{-5}\sx{.8}{$v\!=\!-\!6$}\ero}%<br>
 \put(136, 23){\rot{-5}\sx{.8}{$v\!=\!-\!7$}\ero}%<br>
 \put(137, 14){\rot{-5}\sx{.8}{$v\!=\!-\!8$}\ero}%<br>
 \end{picture} %<br>
 } %<br>

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