File:Exp1mapT200.jpg

Complex map of the first iteration of exponent in the upper half-plane,

$u+\mathrm i v = \mathrm{tet}(1.+\mathrm{ate}(z))=\exp(z)$ in the $z=x+\mathrm i y$ plane.

The additional thick green lines show $u=\Re(L)$ and $n=\Im(L)$ where $L\approx 0.3+1.3 \mathrm i $ is fixed point of logarithm, $L=\ln(L)$.

The generator below, that plots this image, uses representation of the exponential through tetration and arctetration. WIth minimal modification, this code can generate the similar picture for any other iterate of the exponential, even non-integer and eve nomplex iterate.

C++ generator of curves

// Files ado.cin, conto.cin, FSEXP and FSLOG should be loaded in the working directory in order to compile the 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"
#include "fsexp.cin"
#include "fslog.cin"
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
int M=401,M1=M+1;
int N=403,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("exp1map.eps","w"); ado(o,802,402);
fprintf(o,"401 1 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DO(m,M1) X[m]=-4+.02*(m-.5);
DO(n,N1) { y=0.+.01*(n-.5); if(y>Im(Zo)) break; Y[n]=y; }
Y[n]  =Im(Zo)-.00001;
Y[n+1]=Im(Zo)+.00001;
for(m=n+2;m<N1;m++) Y[m]=.01*(m-2-.5);
for(m=-4;m<5;m++){M(m,0) L(m,4)  }
for(n=0;n<5;n++){M(  -4,n) L(4,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=FSEXP(1.+FSLOG(z));
 p=Re(c); q=Im(c);// if(p>-19 && p<19 && ( x<2. ||  fabs(q)>1.e-12 && fabs(p)>1.e-12) )
                  { g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 1 setlinecap\n");   p=2.;q=1;
conto(o,g,w,v,X,Y,M,N, Re(Zo),-p,p);fprintf(o,".03 W 0 1 0 RGB S\n");
conto(o,f,w,v,X,Y,M,N, Im(Zo),-p,p);fprintf(o,".03 W 0 1 0 RGB S\n");
for(m=-8;m<8;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".007 W 0 .6 0 RGB S\n");
for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".007 W .9 0 0 RGB S\n");
for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".007 W 0 0 .9 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-q,q);fprintf(o,".02 W .8 0 0 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-q,q);fprintf(o,".02 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".02 W .5 0 .5 RGB S\n");
for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-q,q);fprintf(o,".02 W 0 0 0 RGB S\n");
fprintf(o,"0 setlinejoin 0 setlinecap\n");
M(Re(Zo),Im(Zo))L(-4,Im(Zo)) fprintf(o,"1 1 1 RGB .022 W S\n");
DO(n,40){M(Re(Zo)-.2*n,Im(Zo))L(Re(Zo)-.2*(n+.4),Im(Zo)) }
 fprintf(o,"0 0 0 RGB .032 W S\n");
fprintf(o,"showpage\n");
fprintf(o,"%c%cTrailer\n",'%','%');
fclose(o);  free(f); free(g); free(w);
      system("epstopdf exp1map.eps"); 
      system(    "open exp1map.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}