File:ExpIte4T.jpg

iterations of exp: $y\!=\!\exp^n(x)$ for various values of $n$.

For non-integer values of $n$, the evaluation is performed using tetration and ArcTetration funcitons,

$y\!=\!\exp^n(x)=\mathrm{tet}\Big(n+\mathrm{ate}(x)\Big)$

C++ generator of curves

// Files ado.cin, fsexp.cin, fslog.cin //should be loaded to 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 "Tania.cin" // need for LambertW
//#include "LambertW.cin" // need for AuZex
//#include "SuZex.cin"
//#include "AuZex.cin"
#include "fsexp.cin"
#include "fslog.cin"

// z_type tra(z_type z){ return exp(z)+z;}
// z_type F(z_type z){ return log(suzex(z));}
// z_type G(z_type z){ return auzex(exp(z));}

#include "ado.cin"
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);

main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;  FILE *o;o=fopen("ExpIte4.eps","w");  ado(o,804,804);
fprintf(o,"402 402 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
for(n=-4;n<5;n++) {M(-4,n)L(4,n)}
for(m=-4;m<5;m++) {M(m,-4)L(m,4)}
// M(M_E,0)L(M_E,1) M(0,M_E)L(1,M_E)  
fprintf(o,".004 W S\n");
// DO(m,700){x=.01 +.02*m; y=Re(LambertW(LambertW(x)));if(m==0) M(x,y) else L(x,y) if(x>12.03||y>12.03) break;} fprintf(o,".033 W 1 0 0 RGB S\n");
// DO(m,700){x=.01 +.02*m; y=Re(LambertW(x));if(m==0) M(x,y) else L(x,y) if(x>12.03||y>12.03) break;} fprintf(o,".04 W 1 .5 0 RGB S\n");
// M(0,0) L(12.03,12.03) fprintf(o,".04 W 0 1 0 RGB S\n");
DO(m,700){x=-4.02+.02*m; y=exp(x);          if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;} fprintf(o,".032 W 0 1 0 RGB S\n");
DO(m,700){x=-4.02+.02*m; y=exp(exp(x));     if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;} fprintf(o,".032 W 0 1 0 RGB S\n");
DO(m,700){x=-4.02+.02*m; y=exp(exp(exp(x)));if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;} fprintf(o,".032 W 0 1 0 RGB S\n");
DO(m,700){y=-4.02+.02*m; x=exp(y);          if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;} fprintf(o,".032 W 1 0 1 RGB S\n");
DO(m,700){y=-4.02+.02*m; x=exp(exp(y));     if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;} fprintf(o,".032 W 1 0 1 RGB S\n");
DO(m,700){y=-4.02+.02*m; x=exp(exp(exp(y)));if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;} fprintf(o,".032 W 1 0 1 RGB S\n");
for(n=0;n<34;n+=1) {DO(m,700){x=-4.01 +.02*m; y=Re(FSLOG(x)); y=Re(FSEXP(.1*n+y)); if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;}}
for(n=-33;n<0;n+=1){t=Re(FSEXP( FSLOG(-4.)-.1*n));
                    DO(m,700){x=t +.02*m; y=Re(FSLOG(x)); y=Re(FSEXP(.1*n+y)); if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;}}
  fprintf(o,".01 W 0 0 0 RGB S\n");
fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%');  fclose(o);
      system("epstopdf ExpIte4.eps"); 
      system(    "open ExpIte4.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

Latex Generator of labels

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References

http://www.ams.org/journals/bull/1993-29-02/S0273-0979-1993-00432-4/S0273-0979-1993-00432-4.pdf Walter Bergweiler. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29 (1993), 151-188.

http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D. Kouznetsov. Solution of F(x+1)=exp(F(x)) in complex z-plane. 78, (2009), 1647-1670

http://www.ils.uec.ac.jp.jp/~dima/PAPERS/2009vladie.pdf (English)
http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf (English)
http://mizugadro.mydns.jp/PAPERS/2009vladir.pdf (Russian version)
D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.

http://reference.wolfram.com/mathematica/ref/Nest.html Nest, Wolfram Mathematica 9 Documentation center, 2013