File:ArcKellerMapT.png

Complex map of function ArcKeller$=\mathrm{Keller}^{-1}$,

$\displaystyle \mathrm{ArcKeller}(z)=z+\ln\!\left( \frac{1}{\mathrm e}+\frac{\mathrm e \!-\!1}{\mathrm e}\, \mathrm e^{-z} \right)$

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.

#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 Shoko(z_type  z) { return log(1.+exp(z)*(M_E-1.)); }
z_type Shoka(z_type  z) { return z + log(exp(-z)+(M_E-1.)); }
// z_type Keller(z_type z) { return z + log(exp(-z)+(M_E-1.));} // The same as Shoka???
// z_type Keller(z_type z) { return log(1.+M_E*(exp(z)-1.));}
// z_type Keller(z_type z) { return z + log(M_E- exp(-z)*(M_E-1.) );}
z_type ArcKeller(z_type z) { return z + log(1./M_E+ exp(-z)*(1.-1./M_E) );}

main(){ int j,k,m,n; DB x,y, p,q, t,r; z_type z,c,d;

 r=log(M_E-1.); printf("%16.4, r=%16.14f\n",M_E-1.,r); 
// r=log(1./(M_E-1.)); printf("r=%16.14f\n",r); 
// 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("ArcKellerMap.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=ArcKeller(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=-M_PI; y<5; 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<85;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<85;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 ArcKellerMap.eps");    
      system(    "open ArcKellerMap.pdf");
printf("r=%16.14f\n",r); 
      getchar(); system("killall Preview");
}

Latex generator of labels

% File ArcKellerMap.pdf should ge benerated with the code above in order to compile the Latex document below.

% % Gerenator of ArcKellerMapT.png %<br> % Copyleft 2012 by Dmitrii Kouznetsov %<br> \documentclass[12pt]{article} %<br> \usepackage{geometry} %<br> \usepackage{graphicx} %<br> \usepackage{rotating} %<br> \paperwidth 854pt %<br> \paperheight 844pt %<br> \topmargin -96pt %<br> \oddsidemargin -98pt %<br> \textwidth 1100pt %<br> \textheight 1100pt %<br> \pagestyle {empty} %<br> \newcommand \sx {\scalebox} %<br> \newcommand \rot {\begin{rotate}} %<br> \newcommand \ero {\end{rotate}} %<br> \newcommand \ing {\includegraphics} %<br> \begin{document} %<br> \sx{5}{ \begin{picture}(164,165) %<br> \put(6,5){\ing{ArcKellerMap}} %<br> \put(2,163){\sx{.7}{$y$}} %<br> \put(2,144){\sx{.6}{$6$}} %<br> \put(2,124){\sx{.6}{$4$}} %<br> \put(2,104){\sx{.6}{$2$}} %<br> \put(3,116){\sx{.6}{$\pi$}} %<br> \put(18,115.6){\sx{.8}{\bf cut}} %<br> %\put(120,116.7){\sx{.4}{\bf cut}} %<br> \put(2, 84){\sx{.6}{$0$}} %<br> \put(-2,64){\sx{.6}{$-2$}} %<br> \put(-2,53){\sx{.6}{$-\pi$}} %<br> \put(18,52.4){\sx{.8}{\bf cut}} %<br> %\put(11,20.4){\sx{.8}{$v\!\approx\! -\!2\pi$}} %<br> \put(-2,44){\sx{.6}{$-4$}} %<br> \put(-2,24){\sx{.6}{$-6$}} %<br> \put( 22,0){\sx{.6}{$-6$}} %<br> \put( 42,0){\sx{.6}{$-4$}} %<br> \put( 62,0){\sx{.6}{$-2$}} %<br> \put( 86,0){\sx{.6}{$0$}} %<br> \put(106,0){\sx{.6}{$2$}} %<br> \put(126,0){\sx{.6}{$4$}} %<br> \put(146,0){\sx{.6}{$6$}} %<br> \put(164,0){\sx{.7}{$x$}} %<br> \put( 67.2, 75){\rot{90}\sx{.5}{$u\!=\!-0.4$}\ero}%<br> \put( 89.8, 77){\rot{90}\sx{.7}{$u\!=\!0$}\ero}%<br> \put(107., 77){\rot{90}\sx{.7}{$u\!=\!1$}\ero}%<br> \put(118.6, 77){\rot{90}\sx{.7}{$u\!=\!2$}\ero}%<br> \put(129.2, 77){\rot{90}\sx{.7}{$u\!=\!3$}\ero}%<br> \put(132,154.4){\sx{.8}{$v\!=\!7$}}%<br> \put(132,144.3){\sx{.8}{$v\!=\!6$}}%<br> \put(132,134.3){\sx{.8}{$v\!=\!5$}}%<br> \put(132,124.2){\sx{.8}{$v\!=\!4$}}%<br> \put(132,114.2){\sx{.8}{$v\!=\!3$}}%<br> \put(132,104.2){\sx{.8}{$v\!=\!2$}}%<br> \put(132, 94.2){\sx{.8}{$v\!=\!1$}}%<br> \put(132, 84){\sx{.8}{$v\!=\!0$}}%<br> \put(132, 73.9){\sx{.8}{$v\!=\!-\!1$}}%<br> \put(132, 63.4){\sx{.8}{$v\!=\!-\!2$}}%<br> \put(132, 53.4){\sx{.8}{$v\!=\!-\!3$}}%<br> \put(132, 43.4){\sx{.8}{$v\!=\!-\!4$}}%<br> \put(132, 33.4){\sx{.8}{$v\!=\!-\!5$}}%<br> \put(132, 23.4){\sx{.8}{$v\!=\!-\!6$}}%<br> \put(132, 13.4){\sx{.8}{$v\!=\!-\!7$}}%<br> \end{picture} %<br> } %<br> \end{document} %<br> %