File:ExpitploT.png

Explicit plot of iterates of exponential, \(y=\exp^n(x)\).

Similar plot appears
in book «Суперфункции» ^[1], стр. 249, рис.17.3, at the top, and
in book «Superfunctions» ^[2], page 243, Fig.17.3, at the top.

The non-integer iterates are exp[ressed through the Natural tetration tet and the ArcTetration ate: \[ \exp^n(z)=\mathrm{tet}\big( n + \mathrm{ate}(z)\big) \]

The thick blue curve corresponds to the special case \(n=1/2\); \(\varphi=\exp^{1/2}\) ;   \(\varphi(\varphi(z))=\exp(z)\) . This case had been announced by Hellmuth Kneser ^[3], 1950; the efficient algorithms for the evaluation ^[4][5] had been reported in 2009 and in 2010.

C++ generator of curves

/* files fsexp.cin, fslog.cin, ado.cin should be loaded */

#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 "fsexp.cin"
#include "fslog.cin"
#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);
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;  
FILE *o;o=fopen("expitplo.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=-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");
M(-4.01,-4.01) L(4.01,4.01) fprintf(o,".032 W 0 0 1 RGB S\n");

DO(m,700){x=-4.01 +.02*m; y=Re(FSLOG(x)); y=Re(FSEXP(.5+y)); if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;}
fprintf(o,".04 W 0 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 expiteplot.eps"); 
      system("epstopdf expitplo.eps"); 
      system(    "open expitplo.pdf"); //for macintosh
}

Latex generator of labels

\documentclass[12pt]{article} 
\usepackage{geometry} 
\usepackage{graphicx} 
\usepackage{rotating} 
\paperwidth 806pt 
\paperheight 806pt 
\topmargin -105pt 
\oddsidemargin -73pt 
\textwidth 1100pt 
\textheight 1100pt 
\pagestyle {empty} 
\newcommand \sx {\scalebox} 
\newcommand \rot {\begin{rotate}} 
\newcommand \ero {\end{rotate}} 
\newcommand \ing {\includegraphics} 
\parindent 0pt% <br> 
\pagestyle{empty} 
\begin{document} 
\begin{picture}(802,802) 
%\put(10,10){\ing{PowPlo}} 
%\put(0,0){\ing{TraItu3}} 
%\put(0,0){\ing{ExpIte4}} 
%\put(0,0){\ing{expitplo}} 
\put(0,0){\ing{expitplo}} 
\put(411,788){\sx{3}{$y$}} 
\put(411,693.6){\sx{2.9}{$3$}} 
\put(411,593.4){\sx{2.9}{$2$}} 
\put(411,493.2){\sx{2.9}{$1$}} 
\put(411,393){\sx{2.9}{$0$}} 
\put(407,292.8){\sx{2.9}{$-1$}} 
\put(407,192.6){\sx{2.9}{$-2$}} 
\put(407,092.4){\sx{2.9}{$-3$}} 
 
\put(081,408){\sx{2.9}{$-3$}} 
\put(181,408){\sx{2.9}{$-2$}} 
\put(281,408){\sx{2.9}{$-1$}} 
\put(396,408){\sx{2.9}{$0$}} 
\put(497,408){\sx{2.9}{$1$}} 
\put(597,408){\sx{2.9}{$2$}} 
\put(697,408){\sx{2.9}{$3$}} 
\put(787,408){\sx{3}{$x$}} 
 
\put(6,748){\sx{3}{\rot{4}$n\!=\!3.2$\ero}} 
\put(6,708){\sx{3}{\rot{3}$n\!=\!3.1$\ero}} 
\put(6,675){\sx{3}{\rot{3}$n\!=\!3$\ero}} 
\put(5,643){\sx{3}{\rot{3}$n\!=\!2.9$\ero}} 
\put(6,345){\sx{3}{\rot{1}$n\!=\!0.6$\ero}} 
%
\put(7,308){\sx{3}{\rot{4}$n\!=\!0.4$\ero}} 
\put(7,280){\sx{3}{\rot{5}$n\!=\!0.3$\ero}} 
\put(8,242){\sx{3}{\rot{9}$n\!=\!0.2$\ero}} 
\put(9,185){\sx{3}{\rot{11}$n\!=\!0.1$\ero}} 
 
\put(50,36){\sx{3}{\rot{45}$n\!=\!0$\ero}} 
% <br> 
\put(202,5){\sx{3}{\rot{76}$n\!=\!-0.1$\ero}} 
\put(263,5){\sx{3}{\rot{82}$n\!=\!-0.2$\ero}} 
\put(299,5){\sx{3}{\rot{84}$n\!=\!-0.3$\ero}} 
\put(691,5){\sx{3}{\rot{84}$n\!=\!-3$\ero}} 
\put(724,5){\sx{3}{\rot{83}$n\!=\!-3.1$\ero}} 
\put(764,5){\sx{3}{\rot{82}$n\!=\!-3.2$\ero}} 
 
\put(484,600){\sx{3.6}{\rot{69}$y\!=\!\exp(x)$\ero}} 
\put(533,590){\sx{3.6}{\rot{62}$y\!=\!\exp^{1/2}(x)$\ero}} 
\put(641,630){\sx{3.4}{\rot{44}$y\!=\!x$\ero}} 
\put(650,484){\sx{3.6}{\rot{19}$y\!=\!\ln(x)$\ero}} 
\end{picture} 
\end{document} 

References

Keywords

«Abelfunction», «ArcTetration», «ado.cin», «С++», «Exponential», «Fsexp.cin», «Fslog.cin», «Hellmuth Kneser», «Iterate», «Natural tetration», «Square root of exponential», «Superfunction», «Superfunctions», «Tetration»,

«Суперфункции»,