File:Tetma.jpg

Complex map of the natural tetration without labels. It is used at the front cover of book «Superfunctions»^[1], 2020.

The labeled version of the central part of this map (with axes marked and levels specified) appears also as Fig.14.12 at page 203 of the same book.

The map is used also at the from cover of the Russian version «Суперфункции» ^[2], 2014; the labeled map appears as Рис.14.12 at page 204 of the Russian version.

Levels of constant real part \(u\) of the natural tetration and Levels of constant imaginary part \(v\) of the natural tetration are drawn in the \(x,y\) plane; \[u+\mathrm i v = \mathrm{tet}(x+\mathrm iy)\]

The generator of the map is copipasted below.

C++ generator

/* Files ado.cin, conto.cin, fsexp.cin, fslog.cin should be loaded 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 std::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"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
int M=801,M1=M+1;
int N=401,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("b271t0.eps","w");  ado(o,87,87);
FILE *o;o=fopen("tetma.eps","w");  ado(o,1604,804);
fprintf(o,"802 402 translate\n 100 100 scale\n");
DO(m,M1) X[m]=-8.+.02*(m-.5);
DO(n,N1) Y[n]=-4.+.02*(n-.5);
for(m=-8;m<17;m++) {M(m,-4)L(m,4)}
for(n=-4;n<5;n++) {M(  -8,n)L(8,n)} fprintf(o,"2 setlinecap .006 W 0 0 0 RGB S\n");
//fprintf(o,"/adobe-Roman findfont .6 scalefont setfont\n");
DO(m,M1)DO(n,N1){      g[m*N1+n]=9999;
                       f[m*N1+n]=9999;}
//DB b=sqrt(2);
DO(m,M1){x=X[m]; printf("x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y); if(abs(z+2.)>.04)
       { c=FSEXP(z);
//       c=FSLOG(z);
         p=Re(c); q=Im(c);
         if(p>-9999 && p<9999 && fabs(q)>1.e-12) g[m*N1+n]=p;
         if(q>-9999 && q<9999 && fabs(q)>1.e-12) f[m*N1+n]=q;
       }
        }}
fprintf(o,"1 setlinejoin 2 setlinecap\n");
p=2.; q=1.1;;
//#include "plofu.cin"
p=2;q=1;
for(m=-19;m<19;m++)for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".02 W 0 1 0 RGB S\n");
for(m=0;m<29;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);
                                                fprintf(o,".01 W 1 0 0 RGB S\n");
for(m=0;m<29;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".01 W 0 0 1 RGB S\n");

for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".03 W 1 0 0 RGB S\n");
for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".03 W 0 0 1 RGB S\n");
                conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".03 W .8 0 .8 RGB S\n");
for(m=-31;m<32;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n");

fprintf(o,"0 setlinejoin 0 setlinecap\n");
fprintf(o,"showpage\n");
fprintf(o,"%cTrailer\n",'%');
fclose(o);
       system("epstopdf tetma.eps"); 
       system(    "open tetma.pdf"); //for macintosh
       getchar(); system("killall Preview"); // For macintosh

References

Keywords

«Complex map», «Natural tetration», «Book», «BookMap», «Lambert Academic Publishing», «Superfunction», «Superfunctions», «Tetration», «ado.cin», «conto.cin», «fsexp.cin»,