File:KellerMapT.png

Complex map of the Keller function, $\mathrm{Keller}(z)=z+\ln\!\Big(\mathrm e-(\mathrm e\!-\!1) \mathrm e^{-z}\Big)$.


 * $ u\!+\!\mathrm i v = \mathrm{Keller}(x\!+\! \mathrm i y)$

C++ generator of curves
// Files ado.cin and conto.cin should be loaded in the working directory in order to compile 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<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"

z_type Shoka(z_type z) { return z + log(exp(-z)+(M_E-1.)); } z_type Keller(z_type z) { return z + log(M_E- exp(-z)*(M_E-1.) );}

main{ int j,k,m,n; DB x,y, p,q, t,r; z_type z,c,d; r=log(1-1./M_E); printf("r=%16.14f\n",r); int M=400,M1=M+1; int N=801,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("KellerMap.eps","w");ado(o,162,162); fprintf(o,"81 81 translate\n 10 10 scale 2 setlinecap\n "); DO(m,M1){ t=(m-200)/200.; X[m]=4.005*t*(.5+1.5*t*t);} DO(n,N1)Y[n]=-8.+.02*(n-.5); for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}} for(n=-8;n<9;n++){    M(  -8,n)L(8,n)} fprintf(o,".008 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=Tania(z); p=Re(c);q=Im(c); c=Keller(z); p=Re(c);q=Im(c); if(p>-19. && p<19. && q>-19. && q<19. ){ g[m*N1+n]=p;f[m*N1+n]=q;} }} fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1.4;q=.5; for(m=-10;m<10;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<10;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<10;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"); for(y=-2*M_PI; y<7; y+=2*M_PI) {M(r,y)L(-8.1,y) fprintf(o,"0 setlinecap .04 W 1 1 1 RGB S\n"); for(m=0;m<75;m+=4) {x=r-.04-.1*m; M(x,y) L(x-.12,y)} fprintf(o,".06 W 1 .5 0 RGB S\n"); for(m=2;m<75;m+=4) {x=r-.04-.1*m; M(x,y) L(x-.12,y)} fprintf(o,".06 W 0 .5 1 RGB S\n"); } fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf KellerMap.eps"); system(   "open KellerMap.pdf"); printf("r=%16.14f\n",r); getchar; system("killall Preview"); }

Latex source of labels
% File KellerMap.pdf should be generated with the code above in order to compile the Latex document below.

%  % 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{KellerMap}} % \put(2,163){\sx{.7}{$y$}} % \put(2,144){\sx{.6}{$6$}} % \put(2,124){\sx{.6}{$4$}} % \put(2,104){\sx{.6}{$2$}} % % \put(3,116){\sx{.6}{$\pi$}} % \put(18,147){\sx{.8}{\bf cut}} % \put(2, 84){\sx{.6}{$0$}} % \put(18,84){\sx{.8}{\bf cut}} % \put(-2,64){\sx{.6}{$-2$}} % \put(18,20.8){\sx{.8}{\bf cut}} % \put(-2,44){\sx{.6}{$-4$}} % \put(-2,24){\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( 79.5, 109){\rot{90}\sx{.7}{$u\!=\!1$}\ero}% \put( 97.0, 108){\rot{90}\sx{.7}{$u\!=\!2$}\ero}% \put(108.8, 108){\rot{90}\sx{.7}{$u\!=\!3$}\ero}% \put(119.4, 108){\rot{90}\sx{.7}{$u\!=\!4$}\ero}% \put(129.6, 108){\rot{90}\sx{.7}{$u\!=\!5$}\ero}% \put(132,154.4){\sx{.8}{$v\!=\!7$}}% \put(132,144.3){\sx{.8}{$v\!=\!6$}}% \put(132,134.3){\sx{.8}{$v\!=\!5$}}% \put(132,124.2){\sx{.8}{$v\!=\!4$}}% \put(132,114.2){\sx{.8}{$v\!=\!3$}}% \put(132,104.2){\sx{.8}{$v\!=\!2$}}% \put(132, 94.2){\sx{.8}{$v\!=\!1$}}% \put(132, 84){\sx{.8}{$v\!=\!0$}}% \put(132, 73.9){\sx{.8}{$v\!=\!-\!1$}}% \put(132, 63.4){\sx{.8}{$v\!=\!-\!2$}}% \put(132, 53.4){\sx{.8}{$v\!=\!-\!3$}}% \put(132, 43.4){\sx{.8}{$v\!=\!-\!4$}}% \put(132, 33.4){\sx{.8}{$v\!=\!-\!5$}}% \put(132, 23.4){\sx{.8}{$v\!=\!-\!6$}}% \put(132, 13.4){\sx{.8}{$v\!=\!-\!7$}}% \end{picture} % } % \end{document} % %

Copyleft
Copyleft 2012 by Dmitrii Kouznetsov. The image above and its generators can be used for free; attribute the the source.