File:Exp1exp2t.jpg

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Iterate of exponent (thin black lines) compared to the linar combination of $\exp$ and $\exp^2$, deawn with thick green lines.

Iterates of exponent are evaluated through tetration tet and arctetration ate:

$y=\exp^n(x)=\mathrm{tet}(n+\mathrm{ate}(x))$ , thin lines

The corresponding linear combinations of the first and the second iterates of the exponent are defined with

$y=(2\!-\!n)\exp(x)+(n\!-\!1)\exp(\exp(x))$ , and marked with thick green lines.

Description

This image is modification of figure 15.4 from the book Суперфункции, 2014 [1]

The English translation Superfunctions in 2015 yet is in preparation [2]


Evaluation of the natural tetration tet and the arctetration ate is described also in the Mathematics of computation (2009) and in the Vladicavkaz mathematical journal (2010) [3][4].

References

  1. ↑ https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
    http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
    http://mizugadro.mydns.jp/BOOK/202.pdf Дмитрий Кузнецов. Суперфункции. Lambert Academic Publishing, 2014.
  2. ↑ http://www.ils.uec.ac.jp/~dima/BOOK/437.pdf
    http://mizugadro.mydns.jp/BOOK/437.pdf Dmitrii Kouznetsov. Superfunctions. 2015, In preparation.
  3. ↑ 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/2009analuxp.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
  4. ↑ http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.

C++ generator of curves

Files ado.cin, fsexp.cin, fslog.cin should be loaded into 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 std::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);
int 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);
FILE *o;o=fopen("exp1exp2.eps","w"); ado(o,604,404);
fprintf(o,"402 2 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
for(n=0;n<5;n++) {M(-4,n)L(2,n)}
for(m=-4;m<3;m++) {M(m,0)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(n,11){
DO(m,700){x=-4.02+.02*m; t=exp(x); y=.1*n*exp(t)+(1.-.1*n)*t; if(m==0) M(x,y) else L(x,y) if(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=10;n<21;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 exp1exp2.eps");
      system( "open exp1exp2.pdf"); //for macintosh
// getchar(); system("killall Preview"); // For macintosh
return 0;
}

Latex generator of labels


\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphicx}
\usepackage{rotating}
\paperwidth 616pt
\paperheight 422pt
\topmargin -108pt
\oddsidemargin -72pt
\textwidth 1100pt
\textheight 1100pt
\pagestyle {empty}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\parindent 0pt
\pagestyle{empty}
\begin{document}
\begin{picture}(614,414)
%\put(0,0){\ing{ExpIte4}}
\put(10,10){\ing{exp1exp2}}
\put(1,400){\sx{1.8}{$y = \exp^n(x)$, thin lines}}
\put(1,376){\sx{1.8}{and $(2\!-\!n)\exp(x)+(n\!-\!1)\exp(\exp(x))$,}}
\put(270,350){\sx{1.8}{thick lines}}
\put(-1,306.6){\sx{2.}{$3$}}
\put(-1,206.4){\sx{2.}{$2$}}
\put(-1,106){\sx{2.}{$1$}}
\put(-1, 6){\sx{2.}{$0$}}

\put(094,-6){\sx{2}{$-3$}}
\put(194,-6){\sx{2}{$-2$}}
\put(294,-6){\sx{2}{$-1$}}
\put(409,-6){\sx{2}{$0$}}
\put(509,-6){\sx{2}{$1$}}
\put(600,-5){\sx{2}{$x$}}
%
\put(20,118){\sx{1.8}{\rot{0}$n\!=\!2$\ero}}
\put(20,99){\sx{1.8}{\rot{0}$n\!=\!1.9$\ero}}
%\put(20,88){\sx{1.8}{\rot{0}$n\!=\!1.8$\ero}}
\put(20, 79){\sx{1.8}{\rot{0}$n\!=\!1.7$\ero}}
\put(20, 58){\sx{1.8}{\rot{0}$n\!=\!1.5$\ero}}
\put(20, 39){\sx{1.8}{\rot{0}$n\!=\!1.3$\ero}}
%\put(20, 29){\sx{1.8}{\rot{0}$n\!=\!1.2$\ero}}
\put(20, 19){\sx{1.8}{\rot{0}$n\!=\!1.1$\ero}}
\put(20, -1){\sx{1.8}{\rot{0}$n\!=\!1$\ero}}
%
\put(398,260){\sx{2}{\rot{74}$y\!=\!\exp(\exp(x))$\ero}}
\put(504,220){\sx{2}{\rot{66}$y\!=\!\exp(x)$\ero}}

\end{picture}
\end{document}

Remarks

This plot is generated by request by Ю.

File history

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Date/TimeThumbnailDimensionsUserComment
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