File:E1eplot8.png

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Explicit plot of the two superexponentials to base \(\ \eta=\exp(1/\mathrm e)\)

Description

Superexponential to base \(b\) is solution \(F\) of the transfer equation

\[ F(z+1)=\exp_b(F(z)) \]

For \(b\!=\!\eta\!=\!\exp(1/\mathrm e)\ \), the solutions that have specific asymptotic behavior are considered, they approach the fixed point \(\ \mathrm e \!\approx\! 2.71\ \) ; these solutions are described in Mathematics of Computation, 2012 [1] and in book about Superfunctions; both the Russian version «Суперфункции», 2014 [2] and the English version «Superfunctions», 2020.

Use of this image

Figure 1 of the article COMPUTATION OF THE TWO REGULAR SUPER-EXPONENTIALS TO BASE EXP(1/E), 2012 [1].

Figure 22.4 at page 138 of book «Суперфункции» about Superfunctions (Russian version, 2014) [2]

Figure 10.5 at page 127 pf book «Superfunctions» (English version, 2020) [3][4]

C++ generator of curves

#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 "e1etf.cin"
#include "e1egf.cin"
#include "e1eti.cin"
void ado(FILE *O, int X, int Y)
{       fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%');
        fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y);
        fprintf(O,"/M {moveto} bind def\n");
        fprintf(O,"/L {lineto} bind def\n");
        fprintf(O,"/S {stroke} bind def\n");
        fprintf(O,"/s {show newpath} bind def\n");
        fprintf(O,"/C {closepath} bind def\n");
        fprintf(O,"/F {fill} bind def\n");
        fprintf(O,"/o {.1 0 360 arc C S} bind def\n");
        fprintf(O,"/times-Roman findfont 20 scalefont setfont\n");
        fprintf(O,"/W {setlinewidth} bind def\n");
        fprintf(O,"/RGB {setrgbcolor} bind def\n");}
//#include "ado.cin"
#define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);
#define o(x,y) fprintf(o,"%6.3f %6.3f o\n",0.+x,0.+y);
int main(){ int j,k,m,n; DB p,q,t1,t3,u,v,w,x,y; z_type z,c,d;
//FILE *o;o=fopen("fige1e1.eps","w");ado(o,234,124);
FILE *o;o=fopen("e1eplot.eps","w");ado(o,234,124);
fprintf(o,"32 22 translate\n 10 10 scale\n");
for(m=-3;m<21;m++){if(m!=0){M(m,-2)L(m,10)}} 
for(n= -2;n<11;n++){if(n!=0){M(-3,n)L(20,n)}}   fprintf(o,".006 W 0 0 0 RGB S\n");
M(-3,0)L(20.2,0) M(0, -2)L(0,10.2)              fprintf(o,".03  W 0 0 0 RGB S\n");
M(-3,M_E)L(20.3,M_E)                            fprintf(o,".05  W 0 .8 0 RGB S\n");


fprintf(o,"1 setlinejoin 1 setlinecap\n");

DO(m,227){x=-3+.1*m; y=Re(E1EGF(x));    if(m==0)M(x,y)else L(x,y)} fprintf(o,".05 W 0 0 1 RGB S\n");

//DO(m,220){x=-1.5+.1*m; y=Re(E1ETF(x));        if(m==0)M(x,y)else L(x,y)} fprintf(o,".05 W 1 0 0 RGB S\n");
DO(m,221){x=-1.66+.1*m; y=Re(E1ETF(x)); if(m==0)M(x,y)else L(x,y)} fprintf(o,".05 W 1 0 0 RGB S\n");

DO(n,61){y=-2+.2*n; z=z_type(y,M_E*M_PI); x=Re(E1ETI(z)); 
                        printf("%9.4f %9.4f\n",x,y);
                        if(n==0)M(x,y)else L(x,y)} fprintf(o,".05 W .7 0 .7 RGB S\n");
fprintf(o,".03 W 0 0 0 RGB\n");
t1=-2.7982482;

/*
FILE *i;
i=fopen("walket1.txt","r");
DO(k,30){ j=fscanf(i,"%lf%lf%lf",&u,&v,&w);
         printf("%2d %5.2lf %14.10lf %14.10lf\n",j,u,v,w);
          if(j<3) break;
          if(v>10) break;
          o(v+t1,u)       
        }
printf("\n");

fclose(i);
i=fopen("walket2.txt","r");
DO(k,30){ j=fscanf(i,"%lf%lf%lf%lf",&y,&u,&v,&w);   if(j<4) break;
         printf("%5.2f %5.2lf %14.10lf %14.10lf\n",y,u,v,w);
                o(w+t1,y);
        }

printf("\n");
        t3=20.2874;
i=fopen("walket3.txt","r");
DO(k,30){ j=fscanf(i,"%lf%lf%lf",&u,&v,&w);
         printf("%2d %5.2lf %14.10lf %14.10lf\n",j,u,v,w);
          if(j<3) break;
          if(u>-10)       o(v+t3,u)       
        }
fclose(i);
*/

//fprintf(o,"0 setlinecap\n"); M(-2,0)L(-8,0) fprintf(o,".08 W 1 1 1 RGB S\n");
//DO(m,16){M(-2-.4*(m),0)L(-2-.4*(m+.5),0)} fprintf(o,".09 W 0 0 0 RGB S\n");
//M(-2,0)L(-10,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash  S\n");

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
        system("epstopdf e1eplot.eps");
        system(    "open e1eplot.pdf"); //mac
//      system(    "xpdf e1eplot.pdf"); // linux

//q=1.e-14;
//for(n=-10;n<11;n++){z=q*n; y=Re(E1EGF(z)); 
//printf("E1EGF(%20.17f)=%20.17f\n",q*n,y);}

getchar(); system("killall Preview");// mac
}

Latex generator of labels

\documentclass[12pt]{article} 
%\paperwidth 472px 
%\paperheight 800px 
\paperwidth 232px 
\paperheight 122px 
\textwidth 704px 
\textheight 900px 
\topmargin -111px 
\oddsidemargin -73.4px 
\usepackage{graphics} 
\usepackage{rotating} 
\usepackage[usenames]{color}
\newcommand \sx {\scalebox} 
\newcommand \rot {\begin{rotate}} 
\newcommand \ero {\end{rotate}} 
\newcommand \ing {\includegraphics} 
\newcommand \rmi {\mathrm{i}} 
\parindent 0pt
\pagestyle{empty}
\begin{document} 
\begin{picture}(240,126)
%\put(0,0){\ing{fige1efre}}
%\put(0,0){\ing{fige1ew13}}
\put(0,0){\ing{e1eplot}}
\put( 26,119){\sx{.8}{$y$}}
\put( 27,100){\sx{.7}{$8$}}
\put( 27,80){\sx{.7}{$6$}}
\put( 27,60){\sx{.7}{$4$}}
\put( 27,40){\sx{.7}{$2$}}
\put( 27,20){\sx{.7}{$0$}}
\put( 30,14){\sx{.7}{$0$}}
\put( 50,14){\sx{.7}{$2$}}
\put( 70,14){\sx{.7}{$4$}}
\put( 90,14){\sx{.7}{$6$}}
\put(110,14){\sx{.7}{$8$}}
\put(128,14){\sx{.7}{$10$}}
\put(149,14){\sx{.7}{$12$}}
\put(169,14){\sx{.7}{$14$}}
\put(189,14){\sx{.7}{$16$}}
\put(209,14){\sx{.7}{$18$}}
\put(229,14){\sx{.8}{$x$}}
\put(6,120){\sx{.7}{\rot{-88}$y\!=\!\Re\!\Big(F_{1}(x\!+\!\rmi o)\Big)$\ero}}
%\put(183,95){\sx{1.}{$y\!=\!F_{3}(x)$}}
%\put(193,52){\sx{1.}{$y\!=\!\rme$}}
%\put(193,40){\sx{1.}{$y\!=\!F_{1}(x)$}}  %Dima's new latex does not like this
%\put(183,95){\sx{.8}{$y\!=\!F_{3}(x)$}}
%\put(136,38){\sx{.8}{$y\!=\!\mathrm{SuExp}_{\eta,3}(x)\!=\!F_{1}(x)$}}
%\put(148,95){\sx{.8}{$y\!=\!\mathrm{tet}_\eta(x)\!=\!F_{3}(x)$}}
\put(126,95){\sx{.8}{$y\!=\!\mathrm{SuExp}_{\eta,3}(x)\!=\!F_{3}(x)$}}
\put(148,38){\sx{.8}{$y\!=\!\mathrm{tet}_\eta(x)\!=\!F_{1}(x)$}}
\put(193,52){\sx{.8}{$y\!=\!\mathrm e$}}
\end{picture}
\end{document}

References

  1. ↑ 1.0 1.1 http://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2012-02590-7/S0025-5718-2012-02590-7.pdf
    http://mizugadro.mydns.jp/PAPERS/2012e1eMcom2590.pdf
    H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics of Computation. Math. Comp., v.81 (2012), p. 2207-2227. ISSN 1088-6842(e) ISSN 0025-5718(p)
  2. ↑ 2.0 2.1 https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
    http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Pubilshing, 2014.
  3. ↑ https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics - 2020/7/28
  4. ↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.

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