File:AcomapT200.png

Complex map of function ArcCos.

$f=\arccos(x +\mathrm i y)$ is shown in the $x$, $y$ plane with levels $u=\Re(f)=\mathrm{const}$ and levels $v=\Im(f)=\mathrm{const}$

Thick lines correspond to the integer values.

C++ generator of curves
Files ado.cin and conto.cin should be loaded to the working directory for compilation of the C++ 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=0){return I*log( z + sqrt(z*z-1.) );} else{return I*log( z - sqrt(z*z-1.) );}} if(Re(z)>=0){return -I*log( z + sqrt(z*z-1.) );} else {return -I*log( z - sqrt(z*z-1.) );} }

main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; int M=401,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("acomap.eps","w");ado(o,82,82); fprintf(o,"41 41 translate\n 10 10 scale\n"); //DO(m,M1) X[m]=-4.+.02*(m-.5); DO(n,200)Y[n]=-4.+.02*n; Y[200]=-.002; Y[201]= .002; for(n=202;n-99. && p<99.       &&     q>-99. && q<99        ) {g[m*N1+n]=p; f[m*N1+n]=q; }                      }} // // #include "plofu.cin" fprintf(o,"1 setlinejoin 1 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,".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<11;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".03 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,".03 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".03 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,".03 W 0 0 0 RGB S\n"); // M(-4.02,0)L(-1,0) M(1,0)L(4.02,0) fprintf(o,"0 setlinecap .028 W .9 1 .9 RGB S\n"); DO(m,31){x=-1.-.1*m; M(x,0) L(x-.05,0)} DO(n,31){x= 1.+.1*n; M(x,0) L(x+.05,0)} fprintf(o,".032 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf acomap.eps"); system(   "open acomap.pdf"); getchar; system("killall Preview");//for mac }

Latex generator of labels
\documentclass[12pt]{article} % \paperheight 838px % \paperwidth 844px % \textwidth 1294px % \textheight 1200px % \topmargin -80px % \oddsidemargin -80px % \usepackage{graphics} % \usepackage{rotating} % \newcommand \sx {\scalebox} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \ing {\includegraphics} % \newcommand \rmi {\mathrm{i}} % \begin{document} % \newcommand \zoomax { % \put(16,820){\sx{4.4}{$y$}} % \put(16,630){\sx{4}{$2$}} % \put(16,430){\sx{4}{$0$}} % \put(-4, 230){\sx{4}{$-\!2$}} % \put(220, 5){\sx{4}{$-\!2$}} % \put(443, 5){\sx{4}{$0$}} % \put(643, 5){\sx{4}{$2$}} % \put(831,6){\sx{4}{$x$}} % } % \parindent 0pt % %\sx{8}{\begin{picture}(86,86) \put(0,0){\ing{b271t0}} % \begin{picture}(816,816) % \put(40,30){\sx{10}{\ing{acomap}}} % \zoomax % \put(100,548){\sx{4}{\rot{-20}$u\!=\!2.8$\ero}} % \put(90,478){\sx{4}{\rot{-7}$u\!=\!3$\ero}} % \put(90,381){\sx{4}{\rot{8}$u\!=\!3$\ero}} % \put(312,120){\sx{4}{\rot{66}$u\!=\!2$\ero}} % % \put(540,620){\sx{4}{\rot{66}$u\!=\!1.2$\ero}} % \put(624,586){\sx{4}{\rot{48}$u\!=\!0.8$\ero}} % \put(644,548){\sx{4}{\rot{36}$u\!=\!0.6$\ero}} % \put(660,510){\sx{4}{\rot{22}$u\!=\!0.4$\ero}} % \put(670,472){\sx{4}{\rot{10}$u\!=\!0.2$\ero}} % % \put(90,432){\sx{4}{\bf cut}} % \put(690,432){\sx{4}{\bf cut}} % \put(400,795){\sx{4.1}{$v\!=\!-2$}} % \put(400,550){\sx{4}{$v\!=\!-1$}} % \put(411,431){\sx{3.8}{$v\!=\!0$}} % \put(410,313){\sx{4}{$v\!=\!1$}} % \put(410, 66){\sx{4}{$v\!=\!2$}} % \end{picture} % \end{document} % % %

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There is similar image with wider area available.