Difference between revisions of "File:Z2itmapT.jpg"

Complex map of function \(T(z)=z^2\).

\(\mu+\mathrm i \nu=T(x+\mathrm i y)\)

Superfunctions

This complex map is used as Fig.4.11 at page 43 of book «Superfunctions» ^[1][2]
in order to show the notations and to invite the reader(s) to practice with complex maps.

C++ generator of curves

#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 std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"

DB c=2.;
// z_type F(z_type z) {return (c-1.)/(pow(2.,z)-c);}
z_type F(z_type z) {return pow(z,c);}

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
int M=501,M1=M+1;
int N=501,N1=N+1;
DB X[M1],Y[N1]; DB *g, *f, *w; // w is working array.
g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
char v[M1*N1]; // v is working array
FILE *o;o=fopen("z2itmap.eps","w");  ado(o,1002,1002);
fprintf(o,"501 501 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DO(m,M1) X[m]=-5+.02*(m-.5);
DO(n,N1) Y[n]=-5+.02*(n-.5); 
//for(n=0;n<N1;n++) { Y[n]=1.09*sinh((3./200.)*(n-200)); printf("%3d %9.6f\n",n,Y[n]); }
for(m=-10;m<11;m++){M(m,-5) L(m,5)  }
for(n=-10;n<11;n++){M(  -5,n) L(5,n)}
 fprintf(o,".006 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){      g[m*N1+n]=999;
                       f[m*N1+n]=999;}
DO(m,M1){x=X[m]; if(m/10*10==m) printf("x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y); //if(abs(z+2.)>.019)
  c=F(z);
// p=abs(c-z)/(abs(c)+abs(z)); p=-log(p)/log(10.); if(p>0 && p<17) g[m*N1+n]=p;
 p=Re(c); q=Im(c); if(p>-99 && p<99 ){ g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 1 setlinecap\n");
 p=1.2;q=.4;

for(m=-5;m<5;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<5;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 .8 0 0 RGB S\n");
for(m=0;m<5;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 .8 RGB S\n");
for(m= 1;m<11;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".02 W .8 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,".02 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0.  ),-2*p,2*p); fprintf(o,".02 W .5 0 .5 RGB S\n");
for(m=-10;m<11;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".02 W 0 0 0 RGB S\n");
// fprintf(o,"0 setlinejoin 0 setlinecap\n");
// M(-10,0)L(0,0) fprintf(o,"1 1 1 RGB .02 W S\n");
//#include "plofu.cin"
fprintf(o,"showpage\n");
fprintf(o,"%c%cTrailer\n",'%','%');
fclose(o);  free(f); free(g); free(w);
      system("epstopdf z2itmap.eps"); 
      system(    "open z2itmap.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}
// 

Latex generator of labels

\documentclass[12pt]{article} 
\paperwidth 1044px 
\paperheight 1036px 
\textwidth 1204px 
\textheight 1200px 
\topmargin -86px 
\oddsidemargin -72px 
\usepackage{graphics} 
\usepackage{rotating} 
\newcommand \sx {\scalebox} 
\newcommand \rot {\begin{rotate}} 
\newcommand \ero {\end{rotate}} 
\newcommand \ing {\includegraphics} 
\newcommand \rmi {\mathrm{i}} 
\parindent 0pt
\pagestyle{empty}
\begin{document} 
\parindent 0pt 
\sx{1}{\begin{picture}(1004,1004) 
\put(40,20){\ing{z2itmap}} 
\put(15,1008){\sx{3.3}{$y$}}
\put(15,910){\sx{3.3}{$4$}}
\put(15,810){\sx{3.3}{$3$}}
\put(15,710){\sx{3.3}{$2$}}
\put(15,610){\sx{3.3}{$1$}}
\put(15,510){\sx{3.3}{$0$}}
\put(-8,410){\sx{3.3}{$-1$}}
\put(-8,310){\sx{3.3}{$-2$}}
\put(-8,210){\sx{3.3}{$-3$}}
\put(-8,110){\sx{3.3}{$-4$}}
\put(-8,10){\sx{3.3}{$-5$}}
\put(8,-9){\sx{3.3}{$-5$}}
\put(108,-9){\sx{3.3}{$-4$}}
\put(208,-9){\sx{3.3}{$-3$}}
\put(308,-9){\sx{3.3}{$-2$}}
\put(408,-9){\sx{3.3}{$-1$}}
\put(534,-9){\sx{3.3}{$0$}}
\put(634,-9){\sx{3.3}{$1$}}
\put(734,-9){\sx{3.3}{$2$}}
\put(834,-9){\sx{3.3}{$3$}}
\put(934,-9){\sx{3.3}{$4$}}
\put(1026,-9){\sx{3.3}{$x$}}
\put(889,649){\sx{5}{\rot{-15}$\nu\!=\!10$\ero}}
\put(884,579){\sx{5}{\rot{-7}$\nu\!=\!5$\ero}}
\put(884,510){\sx{5}{$\nu\!=\!0$}}
\put(884,438){\sx{5}{\rot{6}$\nu\!=\!-5$\ero}}
\put(890,338){\sx{5}{\rot{-58}$\mu\!=\!10$\ero}}
\put(838,308){\sx{5}{\rot{-55}$\mu\!=\!5$\ero}}
\put(784,264){\sx{5}{\rot{-45}$\mu\!=\!0$\ero}}
\put(726,214){\sx{5}{\rot{-37}$\mu\!=\!-5$\ero}}
\put(706,154){\sx{5}{\rot{-33}$\mu\!=\!-10$\ero}}
\end{picture}}
\end{document}
%

References

Keywords

«Complex map», «Superfunctions», «Суперфункции»,