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.

#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 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

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

References