File:E1egi4.jpg

Complex map of Abelfunction of the exponential to base \(\eta=\exp(1/\mathrm e)\):

\[ u+\mathrm i y=\mathrm{AuExp}_{\eta, 3}(x\!+\!\mathrm i y) \]

The superexponential \(\ \mathrm{SuExp}_{\eta, 3}\ \) of real argument remains bigger than \(\mathrm e\) ; so, the inverse function \(\ \mathrm{AuExp}_{\eta, 3}\!=\!\mathrm{SuExp}_{\eta, 3}^{~ -1} \ \) has the inevitable cut from \(-\infty\) to \(\mathrm e\) ; it is shown in the map with dashed line.

Descriptions

The growing Abelfunction \(\mathrm{AuExp}_{\eta, 3}\) to "base e1e", est, to base \(\eta=\exp(1/\mathrm e)\) is described in Mathematics of Computation, 2012 ^[1]. The similar map appears as top part of Fig.2 at page 2216.

The similar map is used also as bottom part of Fig.11.6 at page 141 of book «Суперфункции», (Russian version), 2014 ^[2]. Both maps of Fig.11.6 of the Russian version are loaded as http://mizugadro.mydns.jp/t/index.php/File:E1eAuMap600.jpg

In the English version, book «Superfunctions», year 2020 ^[3] the closest analogy is Fig.10.7 at page 130, but the map for abelfunction \(AuExp_{\eta,3}\) appears at the top of the figure as it is described first in the text (the map of the arctetration to base \(\eta\) http://mizugadro.mydns.jp/t/index.php/File:E1eti4.jpg appears at the bottom of that figure).

C++ generator of map

//Files ado.cin , conto.cin , e1egf.cin , e1egi.cin should be loaded 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 "e1egf.cin"
#include "e1egi.cin"
#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
 int M=401,M1=M+1;
 int N=301,N1=N+1;
DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array.
char v[M1*N1]; // v is working array
//E1E Growing Funciton
FILE *o;o=fopen("e1egi.eps","w");ado(o,402,302);
fprintf(o,"101 151 translate\n 10 10 scale\n");
DO(m,M1) X[m]=-10.+.1*(m-.5);
DO(n,N1) Y[n]=-15.+.1*(n-.5);
for(m=-10;m<31;m++){ //if(m==0){        M(m,-10.2)L(m,10.2)} else
                                {       M(m,-15)L(m,15)         }}
for(n=-15;n<16;n++){ M(-10,n)L(30,n)}
fprintf(o,".006 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
DO(m,M1){x=X[m]; printf("run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
        c=E1EGI(z);
//      c=E1EGF(z);
//      d=z;
//      p=abs(c-d)/abs(c+d); p=-log(p)/log(10.);        
        p=Re(c); q=Im(c);
//      if(p>-85 && p<85)  g[m*N1+n]=p;
        if(p>-33 && p<33 && fabs(p)> 1.e-9 && fabs(p-1.)>1.e-9 &&
          q >-33 && q<33 && fabs(q)> 1.e-9) { g[m*N1+n]=p; f[m*N1+n]=q; }
        }}

fprintf(o,"1 setlinejoin 1 setlinecap\n");
p=1.;q=.5;
//#include"plofu.cin"
//p=2;q=1;
for(m=-19;m<19;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".02 W 0 .6 0 RGB S\n");
for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);
                                                fprintf(o,".02 W .9 0 0 RGB S\n");
for(m=0;m<29;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".02 W 0 0 .9 RGB S\n");

for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".07 W .8 0 0 RGB S\n");
for(m= 1;m<20;m++) conto(o,f,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".07 W 0 0 .8 RGB S\n");
                   conto(o,f,w,v,X,Y,M,N,(0.  ),-p,p);fprintf(o,".07 W .5 0 .5 RGB S\n");
for(m=-31;m<32;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".07 W 0 0 0 RGB S\n");

fprintf(o,"0 setlinecap\n");
M(M_E,0)L(-10,0) fprintf(o,".08 W 1 1 1 RGB S\n");
DO(m,36){M(M_E-.4*(m),0)L(M_E-.4*(m+.5),0)} fprintf(o,".09 W 1 0 1 RGB S\n");

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
        system("epstopdf e1egi.eps");
        system(    "open e1egi.pdf"); //mac
//      system(    "xpdf e1egi.pdf"); // linux
getchar(); system("killall Preview");// mac
}

Latex generator of labels

 \documentclass[12pt,a4paper,oneside]{book}
%\newcommand \EN[1] {{#1}}   	% make the English version
\newcommand \EN[1] {{}}    	% suppress the English version
\newcommand \RU[1] {{#1}}   	% make the Russian version (in this document not supported)
%\newcommand \RU[1] {{}}      	% suppress the Russian version
%The Japanese version is not yet supported. While \JP is used to suppress several lines at once.
\newcommand \JP[1] {{}}   	 % ореsuppress some text
%\usepackage[space]{cite }% If exist.
\usepackage[utf8]{inputenc}
\usepackage[T2A]{fontenc}
\usepackage[russian]{babel}
\usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx}

\usepackage{rotating}
\usepackage{hyperref} 
\usepackage{wrapfig}
\usepackage{geometry}
\paperwidth 404pt
\paperheight 300pt
\oddsidemargin -56pt
\topmargin -110pt
\pagestyle{empty}
\large
\usepackage{color}
\definecolor{rose}{rgb}{1,.9,1}
\renewcommand\thesection{\arabic{section}}
\renewcommand\figurename{Рис.}
\newcommand \be {\begin{eqnarray}}
\newcommand \ee {\end{eqnarray} }
\newcommand \sx {\scalebox}
\newcommand \rme {{\rm e}}	 %%makes the base of natural logarithms Roman font 
%\newcommand \rme {{e}}	%%makes the base of natural logarithms Italics font; choose one of these
\newcommand \rmi {{\rm i}}	 %%imaginary unity is always roman font
\newcommand \ds {\displaystyle}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing \includegraphics
\newcommand \tet {\mathrm{tet}}
\begin{document}
\parindent 0pt
\newcommand \figaxe {
\put( -6,298){\sx{1}{$y$}}
%\put( -11,289){\sx{1}{$14$}}
\put( -11,268){\sx{.9}{$12$}}
\put( -11,248){\sx{.9}{$10$}}
\put( -6,228){\sx{.9}{$8$}}
\put( -6,208){\sx{.9}{$6$}}
\put( -6,188){\sx{.9}{$4$}}
\put( -6,168){\sx{.9}{$2$}}
\put( -6, 148){\sx{.9}{$0$}}
\put(-14,128){\sx{.9}{$-2$}}
\put(-14,108){\sx{.9}{$-4$}}
\put(-14,  88){\sx{.9}{$-6$}}
\put(-14,  68){\sx{.9}{$-8$}}
\put(-18.4,  48){\sx{.9}{$-10$}}
\put(-18.4,  28){\sx{.9}{$-12$}}
\put(-18.4,   8){\sx{.9}{$-14$}}
%\put(-13, 79){\sx{1}{$-6$}}
\put(397,-7.5){\sx{1}{$x$}}
\put(377,-8){\sx{.9}{$28$}}
\put(357,-8){\sx{.9}{$26$}}
\put(337,-8){\sx{.9}{$24$}}
\put(317,-8){\sx{.9}{$22$}}
\put(297,-8){\sx{.9}{$20$}}
\put(277,-8){\sx{.9}{$18$}}
\put(257,-8){\sx{.9}{$16$}}
\put(237,-8){\sx{.9}{$14$}}
\put(217,-8){\sx{.9}{$12$}}
\put(197,-8){\sx{.9}{$10$}}
\put(179,-8){\sx{.9}{$8$}}
\put(159,-8){\sx{.9}{$6$}}
\put(139,-8){\sx{.9}{$4$}}
\put(119,-8){\sx{.9}{$2$}}
\put( 99,-8){\sx{.9}{$0$}}
\put( 73,-8){\sx{.9}{$-2$}}
\put( 53,-8){\sx{.9}{$-4$}}
\put( 33,-8){\sx{.9}{$-6$}}
\put( 13,-8){\sx{.9}{$-8$}}
}
%\hskip -28pt
%\vskip 18pt
 \sx{.96}{\begin{picture}(410,310) 
 %\put(0,0){\ing{e1egf} }  % suppress this string to boost
 %\put(0,0){\ing{e1esuma} }  % suppress this string to boost
  \put(0,0){\ing{e1egi} } 
  \normalsize
 \figaxe
 \put(  4,220){\sx{1.2}{$u\!=\!19$}} 
 %\put(166,216){\sx{1.4}{$p\!=\!18$}}
 \put(352,276){\sx{1.2}{$v\!=\!0.2$}}
 \put(350, 26){\sx{1.2}{$v\!=\!-0.2$}}
 \put(350,186){\sx{1.2}{$u\!=\!18.4$}}
 \put(270,176){\sx{1.2}{$u\!=\!18.2$}}
 \put(233,158){\sx{1.2}{$u\!=\!18$}}
 \put(  4, 74){\sx{1.2}{$u\!=\!19$}} 
% \put(166, 80){\sx{1.4}{$p\!=\!18$}}
 \put(20,186){\sx{1.2}{$v\!=\!1$}}
 \put(20,110){\sx{1.2}{$v\!=\!-1$}}
 \put(20,148.4){\sx{1.2}{\bf cut }} 
 \put(377,148.4){\sx{1.1}{$v\!=\!0$}}
 \end{picture}}
\end{document}

References

Keywords

«Abel function», «Abelfunction», «Base e1e», «Exotic iteration», «Fixed point», «Mahtematics of Computation», «Superfunctions», «Superexponential»,

«Суперфункции»,