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

 #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 acos(z_type z){
  if(Im(z)<0){if(Re(z)>=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<N1;n++) Y[n]=-4.+.02*(n-1.);
 DO(m,M1)X[m]=Y[m];
 for(m=-4;m<5;m++){if(m==0){M(m,-4.1)L(m,4.1)} else{M(m,-4)L(m,4)}}
 for(n=-4;n<5;n++){     M(  -4,n)L(4,n)}
 fprintf(o,".01 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=acos(z);
 //     c=ACip5(z);
 //     p=abs((c-d)/(c+d));  p=-log(p)/log(10.);
       p=Re(c);
       q=Im(c);        
       if(p>-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} %<br>
\paperheight 838px %<br>
\paperwidth 844px %<br>
\textwidth 1294px %<br>
\textheight 1200px %<br>
\topmargin -80px %<br>
\oddsidemargin -80px %<br>
\usepackage{graphics} %<br>
\usepackage{rotating} %<br>
\newcommand \sx {\scalebox} %<br>
\newcommand \rot {\begin{rotate}} %<br>
\newcommand \ero {\end{rotate}} %<br>
\newcommand \ing {\includegraphics} %<br>
\newcommand \rmi {\mathrm{i}} %<br>
\begin{document} %<br>
\newcommand \zoomax { %<br>
\put(16,820){\sx{4.4}{$y$}} %<br>
\put(16,630){\sx{4}{$2$}} %<br>
\put(16,430){\sx{4}{$0$}} %<br>
\put(-4, 230){\sx{4}{$-\!2$}} %<br>
\put(220,  5){\sx{4}{$-\!2$}} %<br>
\put(443,  5){\sx{4}{$0$}} %<br>
\put(643,  5){\sx{4}{$2$}} %<br>
\put(831,6){\sx{4}{$x$}}  %<br>
} %<br>
\parindent 0pt %<br>
%\sx{8}{\begin{picture}(86,86) \put(0,0){\ing{b271t0}} %<br>
\begin{picture}(816,816) %<br> 
\put(40,30){\sx{10}{\ing{acomap}}} %<br>
\zoomax %<br>
\put(100,548){\sx{4}{\rot{-20}$u\!=\!2.8$\ero}} %<br> 
\put(90,478){\sx{4}{\rot{-7}$u\!=\!3$\ero}} %<br> 
\put(90,381){\sx{4}{\rot{8}$u\!=\!3$\ero}} %<br> 
\put(312,120){\sx{4}{\rot{66}$u\!=\!2$\ero}} %<br> 
 %<br> 
\put(540,620){\sx{4}{\rot{66}$u\!=\!1.2$\ero}} %<br> 
\put(624,586){\sx{4}{\rot{48}$u\!=\!0.8$\ero}} %<br> 
\put(644,548){\sx{4}{\rot{36}$u\!=\!0.6$\ero}} %<br> 
\put(660,510){\sx{4}{\rot{22}$u\!=\!0.4$\ero}} %<br> 
\put(670,472){\sx{4}{\rot{10}$u\!=\!0.2$\ero}} %<br> 
 %<br> 
\put(90,432){\sx{4}{\bf cut}} %<br> 
\put(690,432){\sx{4}{\bf cut}} %<br> 
\put(400,795){\sx{4.1}{$v\!=\!-2$}} %<br> 
\put(400,550){\sx{4}{$v\!=\!-1$}} %<br> 
\put(411,431){\sx{3.8}{$v\!=\!0$}} %<br> 
\put(410,313){\sx{4}{$v\!=\!1$}} %<br> 
\put(410, 66){\sx{4}{$v\!=\!2$}} %<br> 
\end{picture} %<br>
\end{document} %<br> 
%</nowiki> %<br>

Related files

There is similar image with wider area available.