File:ArcTaniaMap.png

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

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. ;}

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); DO(n,80)Y[n]=-8.+.1*n; Y[80]=-.033; Y[81]= .033; for(n=82;n-99. && p<99. && //      (fabs(y)>.034 ||x>-.9 ||fabs(x-int(x))>1.e-3) &&           q>-99. && q<99 //&& fabs(q)> 1.e-19        ) {g[m*N1+n]=p;f[m*N1+n]=q;} }} fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1.8;q=.7; //p=2;q=1; 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 % % 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{arctaniacontour}} % \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(23,100){\sx{.8}{$u\!=\!0$}} % \put(2, 84){\sx{.6}{$0$}} % \put(59,85){\sx{.6}{\bf cut}} % % \put(20, 84){\sx{.8}{$v\!=\!0$}} % \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( 81, 23){\rot{81}\sx{.8}{$u\!=\!0$}\ero}% \put( 92, 23){\rot{82}\sx{.8}{$u\!=\!1$}\ero}% \put(101, 22){\rot{82}\sx{.8}{$u\!=\!2$}\ero}% \put(111, 21){\rot{83}\sx{.8}{$u\!=\!3$}\ero}% \put(120, 21){\rot{84}\sx{.8}{$u\!=\!4$}\ero}%

\put(139,155){\rot{4}\sx{.8}{$v\!=\!8$}\ero}% \put(138,146){\rot{4}\sx{.8}{$v\!=\!7$}\ero}% \put(138,136){\rot{4}\sx{.8}{$v\!=\!6$}\ero}% \put(138,127){\rot{4}\sx{.8}{$v\!=\!5$}\ero}% \put(137,118){\rot{4}\sx{.8}{$v\!=\!4$}\ero}% \put(136,109){\rot{4}\sx{.8}{$v\!=\!3$}\ero}% \put(135,100){\rot{4}\sx{.8}{$v\!=\!2$}\ero}% \put(134, 92){\rot{3}\sx{.8}{$v\!=\!1$}\ero}% \put(134, 84){\rot{0}\sx{.8}{$v\!=\!0$}\ero}% \put(134, 76){\rot{-3}\sx{.8}{$v\!=\!-\!1$}\ero}% \put(133, 68){\rot{-5}\sx{.8}{$v\!=\!-\!2$}\ero}% \put(134, 59){\rot{-5}\sx{.8}{$v\!=\!-\!3$}\ero}% \put(135, 51){\rot{-5}\sx{.8}{$v\!=\!-\!4$}\ero}% \put(135, 41){\rot{-5}\sx{.8}{$v\!=\!-\!5$}\ero}% \put(135, 32){\rot{-5}\sx{.8}{$v\!=\!-\!6$}\ero}% \put(136, 23){\rot{-5}\sx{.8}{$v\!=\!-\!7$}\ero}% \put(137, 14){\rot{-5}\sx{.8}{$v\!=\!-\!8$}\ero}% \end{picture} % } % \end{document}