File:AcosmapT200.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=400,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("acosmap.eps","w");ado(o,162,162); fprintf(o,"81 81 translate\n 10 10 scale\n"); DO(m,M1) X[m]=-8.+.04*(m-.5); DO(n,200)Y[n]=-8.+.04*n; Y[200]=-.015; Y[201]= .015; for(n=202;n-99. && p<99. &&         q>-99. && q<99 //&& fabs(q)> 1.e-19       ) {g[m*N1+n]=p;f[m*N1+n]=q;} }} 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,".016 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,".016 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,".016 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,".04 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,".04 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".04 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,".04 W 0 0 0 RGB S\n"); fprintf(o,"0 setlinecap\n"); M(-1,0)L(-8,0) M(1,0)L(8,0) fprintf(o,"0.05 W 1 1 1 RGB S\n"); DO(m,35) {x=-1.-.2*m; M(x,0) L(x-.1,0)} DO(m,35) {x= 1.+.2*m; M(x,0) L(x+.1,0)} fprintf(o,".05 W 0 .3 0 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf acosmap.eps"); system(   "open acosmap.pdf"); getchar; system("killall Preview");//for mac }

Latex generator of labels
For compilation of the Latex code below, file acosmap.pdf should be generated with the code from the previous section.

% 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{acosmap}} % \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(4,134){ \sx{.8}{\rot{-30}$u\!=\!2.6$\ero}} % \put(4,114){ \sx{.8}{\rot{-18}$u\!=\!2.8$\ero}} % \put(3,99){ \sx{.8}{\rot{-10}$u\!=\!3$\ero}} % \put(2, 84){\sx{.6}{$0$}} % \put(8, 84){\sx{.8}{\bf cut}} % \put(7,75){\sx{.8}{\rot{10}$u\!=\!3$\ero}} % \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( 54, 161){\rot{-67}\sx{.82}{$u\!=\!2$}\ero}% \put( 69, 165){\rot{-77}\sx{.82}{$u\!=\!1.8$}\ero}% \put( 76, 122){\rot{.1}\sx{.86}{$v\!=\!-2$}\ero}% \put( 76, 45){\rot{-.1}\sx{.86}{$v\!=\!2$}\ero}% \put( 9, 24){\rot{-42}\sx{.8}{$v\!=\!3$}\ero}% \put(137, 108){\rot{23}\sx{.8}{$u\!=\!0.4$}\ero}% \put(137, 96){\rot{15}\sx{.8}{$u\!=\!0.2$}\ero}% \put(144, 84){\rot{0}\sx{.8}{\bf cut}\ero}% \put(140, 76){\rot{-11}\sx{.72}{$u\!=\!0.2$}\ero}% \put(140, 65){\rot{-21}\sx{.72}{$u\!=\!0.4$}\ero}% \put(136, 54){\rot{-32}\sx{.72}{$u\!=\!0.6$}\ero}% \put(128, 46){\rot{-44}\sx{.72}{$u\!=\!0.8$}\ero}% \put(120, 37){\rot{-56}\sx{.8}{$u\!=\!1$}\ero}% \end{picture} % } % \end{document}

Keywords
ArcCos, Complex map