File:Gz2z1mapT.jpg

Complex map of function $g(z)=\log_2(1+1/z)$

$u+\mathrm i v= g(x+\mathrm i y)$

C++ generator of curves
// using namespace std; typedef std::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"

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

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; int M=1001,M1=M+1; int N=1001,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("gz2z1map.eps","w"); ado(o,2002,2002); fprintf(o,"1001 1001 translate\n 100 100 scale\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); DO(m,M1) X[m]=-10+.02*(m-.5); DO(n,N1) Y[n]=-10+.02*(n-.5); //for(n=0;n.019) c=G(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>-19 && p<19 ){ 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=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".014 W 0 .6 0 RGB S\n"); for(m=0;m<4;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".014 W .8 0 0 RGB S\n"); for(m=0;m<4;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".014 W 0 0 .8 RGB S\n"); for(m= 1;m<6;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".05 W .8 0 0 RGB S\n"); for(m= 1;m<6;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".05 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".05 W .5 0 .5 RGB S\n"); for(m=-6;m<7;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".05 W 0 0 0 RGB S\n");

fprintf(o,"0 setlinejoin 0 setlinecap\n"); M(0,0)L(-1,0) fprintf(o,"1 1 1 RGB .02 W S\n"); DO(n,10) {M(-.1*(n),0)L(-.1*(n-.3),0)} fprintf(o,"0 0 0 RGB .025 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 gz2z1map.eps"); system(   "open gz2z1map.pdf"); //for macintosh getchar; system("killall Preview"); // For macintosh } //

Latex generator of labels
% \documentclass[12pt]{article} \paperwidth 2004px \paperheight 2004px \textwidth 2394px \textheight 2300px \topmargin -101px \oddsidemargin -70px \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}(2002,2002) \put(30,1950){\sx{8}{$y$}} \put(30,1780){\sx{7}{$8$}} \put(30,1580){\sx{7}{$6$}} \put(30,1380){\sx{7}{$4$}} \put(30,1180){\sx{7}{$2$}} \put(30,0980){\sx{7}{$0$}} \put(-22,780){\sx{7}{$-2$}} \put(-22,580){\sx{7}{$-4$}} \put(-22,380){\sx{7}{$-6$}} \put(-22,180){\sx{7}{$-8$}} \put(120, 6){\sx{7}{$-8$}} \put(320, 6){\sx{7}{$-6$}} \put(520, 6){\sx{7}{$-4$}} \put(720, 6){\sx{7}{$-2$}} \put(988, 6){\sx{7}{$0$}} \put(1188, 6){\sx{7}{$2$}} \put(1388, 6){\sx{7}{$4$}} \put(1588, 6){\sx{7}{$6$}} \put(1788, 6){\sx{7}{$8$}} \put(1958, 6){\sx{8}{$x$}} \put(0,0){\ing{gz2z1map}} %\put(0,0){\ing{10}} \put(880,1920){\sx{8}{$u\!=\!0$}} \put(210,1888){\sx{8}{$v\!=\!-0.1$}} \put(1610,1888){\sx{8}{$v\!=\!-0.1$}}

\put(250,980){\sx{8}{$v\!=\!0$}} \put(1740,980){\sx{8}{$v\!=\!0$}}

\put(80,850){\sx{8}{$u\!=\!-0.2$}} \put(1600,860){\sx{8}{$u\!=\!0.2$}}

\put(1040,250){\sx{8}{$v\!=\!0.2$}} \put(250,232){\sx{8}{$u\!=\!-0.1$}} \put(1740,260){\sx{8}{$u\!=\!0.1$}} \put(210,110){\sx{8}{$v\!=\!0.1$}} \put(1610,110){\sx{8}{$v\!=\!0.1$}}

\put(880,120){\sx{8}{$u\!=\!0$}} \end{picture}} \end{document} %