Difference between revisions of "File:ExpIte4T.jpg"

Figure 15.4 from page 210 of book «Superfunctions» ^[1], 2020.

The same plot appears also as Рис.15.5 at page 212 of the Russian version «Суперфункции»^[2], 2014.

The plot shows iterates of exponential: \(y\!=\!\exp^n(x)\) for various values of \(n\).

For non-integer values of \(n\), the evaluation is performed using the Natural tetration tet and the ArcTetration ate functions,

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

In the next section, the C++ implementations of functions tet and ate are denoted with identifiers FSEXP and FSLOG.

For \(n=0.5\) the square root of exponential \(\varphi=\exp^{1/2}=\sqrt{\exp}\) satisfies equation

\(\varphi(\varphi(z))=\mathrm e^z\)

This function had been announced in 1950 by Hellmuth Kneser ^[3], but only in 2009 an 2010 the efficient algorithms ^[4][5] for the evaluation had been implemented and published.

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

<br>
% file ExpIte4.pdf should be generated with the code above in order to compile the [[Latex]] document below. %<br>
% Copyleft 2012 by Dmitrii Kouznetsov <br> %
\documentclass[12pt]{article}  % <br> 
\usepackage{geometry}  % <br> 
\usepackage{graphicx} % <br> 
\usepackage{rotating} % <br> 
\paperwidth 806pt % <br> 
\paperheight 806pt % <br> 
\topmargin -105pt % <br> 
\oddsidemargin -73pt % <br> 
\textwidth 1100pt % <br> 
\textheight 1100pt % <br> 
\pagestyle {empty} % <br> 
\newcommand \sx {\scalebox} % <br> 
\newcommand \rot {\begin{rotate}} % <br> 
\newcommand \ero {\end{rotate}} % <br> 
\newcommand \ing {\includegraphics} % <br> 
\parindent 0pt%  <br> 
\pagestyle{empty} % <br> 
\begin{document}  % <br> 
\begin{picture}(802,802) % <br> 
%\put(10,10){\ing{PowPlo}} % <br> 
%\put(0,0){\ing{TraItu3}} % <br> 
\put(0,0){\ing{ExpIte4}} % <br> 
\put(411,788){\sx{3}{$y$}} % <br> 
\put(411,693.6){\sx{2.9}{$3$}} % <br> 
\put(411,593.4){\sx{2.9}{$2$}} % <br> 
\put(411,493.2){\sx{2.9}{$1$}} % <br> 
\put(411,393){\sx{2.9}{$0$}} % <br> 
\put(407,292.8){\sx{2.9}{$-1$}} % <br> 
\put(407,192.6){\sx{2.9}{$-2$}} % <br> 
\put(407,092.4){\sx{2.9}{$-3$}} % <br> 
 % <br> 
\put(081,408){\sx{2.9}{$-3$}} % <br> 
\put(181,408){\sx{2.9}{$-2$}} % <br> 
\put(281,408){\sx{2.9}{$-1$}} % <br> 
\put(396,408){\sx{2.9}{$0$}} % <br> 
\put(497,408){\sx{2.9}{$1$}} % <br> 
\put(597,408){\sx{2.9}{$2$}} % <br> 
\put(697,408){\sx{2.9}{$3$}} % <br> 
\put(787,408){\sx{3}{$x$}} % <br> 
 % <br> 
\put(6,748){\sx{3}{\rot{4}$n\!=\!3.2$\ero}} % <br> 
\put(6,708){\sx{3}{\rot{3}$n\!=\!3.1$\ero}} % <br> 
\put(6,675){\sx{3}{\rot{3}$n\!=\!3$\ero}} % <br> 
\put(5,643){\sx{3}{\rot{3}$n\!=\!2.9$\ero}} % <br> 
\put(6,345){\sx{3}{\rot{1}$n\!=\!0.6$\ero}} % <br> 
%
\put(7,308){\sx{3}{\rot{4}$n\!=\!0.4$\ero}} % <br> 
\put(7,280){\sx{3}{\rot{5}$n\!=\!0.3$\ero}} % <br> 
\put(8,242){\sx{3}{\rot{9}$n\!=\!0.2$\ero}} % <br> 
\put(9,185){\sx{3}{\rot{11}$n\!=\!0.1$\ero}} % <br> 
 % <br> 
\put(50,36){\sx{3}{\rot{45}$n\!=\!0$\ero}} % <br> 
% <br> 
\put(202,5){\sx{3}{\rot{76}$n\!=\!-0.1$\ero}} % <br> 
\put(263,5){\sx{3}{\rot{82}$n\!=\!-0.2$\ero}} % <br> 
\put(299,5){\sx{3}{\rot{84}$n\!=\!-0.3$\ero}} % <br> 
\put(691,5){\sx{3}{\rot{84}$n\!=\!-3$\ero}} % <br> 
\put(724,5){\sx{3}{\rot{83}$n\!=\!-3.1$\ero}} % <br> 
\put(764,5){\sx{3}{\rot{82}$n\!=\!-3.2$\ero}} % <br> 
 % <br> 
\put(480,600){\sx{3.6}{\rot{69}$y\!=\!\exp(x)$\ero}} % <br> 
\put(641,630){\sx{3.4}{\rot{44}$y\!=\!x$\ero}} % <br> 
\put(650,484){\sx{3.6}{\rot{19}$y\!=\!\ln(x)$\ero}} % <br> 

\end{picture} % <br> 
\end{document} % <br> 
%

References

Keywords

«Abelfunction», «Arctetration», «Exponential», «Natural tetration», «Iterate of exponential», «Iteration», «Superfunction», «Superfunctions», «Tetration», «ado.cin», «fsexp.cin», «fslog.cin»,