File:E1etetma8.jpg

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Original file(4,472 × 3,320 pixels, file size: 1.69 MB, MIME type: image/jpeg)

Complex map of tetration to base e1e

$u\!+\!\mathrm i v = \mathrm{tet}_{\eta}(x\!+\!\mathrm i y)$

$\eta\!=\!\exp(1/\mathrm e)\!\approx\! 1.444667861$

Tetration $F=\mathrm{tet}_{\eta}$ is real-holomorphic solution of equation

$\exp_\eta(F(z))=F(z+1)$

with specific value at zero, $F(0)\!=\!1$, and specific asymptotic behaviour,

Use of this image

Similar map is used in figure 11.5 of the book Суперфункции (in Russian) [1]

and in the article [2]

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 Д.Кузнецов. Суперфункции. Lambert Academic Press, 2014. Рисунок 11.5
  2. http://www.ams.org/journals/mcom/2012-81-280/S0025-5718-2012-02590-7/home.html
    http://www.ils.uec.ac.jp/~dima/PAPERS/2012e1eMcom2590.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, v.81 (2012), 2207-2227.

C++ generator of map


#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 "e1etf.cin"
#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
 int M=501,M1=M+1;
 int N=601,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 Growung Function
//FILE *o;o=fopen("e1etf.eps","w");ado(o,402,302);
FILE *o;o=fopen("e1etetma.eps","w");ado(o,402,302);
fprintf(o,"101 151 translate\n 10 10 scale\n");
DO(m,M1) X[m]=-10.+.08*(m-.5);
DO(n,N1) Y[n]=-15.+.05*(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,"2 setlinecap .02 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=E1ETF(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-8 && fabs(p-1.)>1.e-8 &&
          q >-33 && q<33 && fabs(q)> 1.e-8) { g[m*N1+n]=p; f[m*N1+n]=q; }
        }}

fprintf(o,"1 setlinejoin 1 setlinecap\n");
p=1.8;q=1.;
//#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,".03 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,".03 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,".03 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,".09 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,".09 W 0 0 .8 RGB S\n");
                   conto(o,f,w,v,X,Y,M,N,(0. ),-p,p);fprintf(o,".09 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,".09 W 0 0 0 RGB S\n");

fprintf(o,"0 setlinecap\n");
// M(-2.,0)L(-10.1,0) fprintf(o,"0 setlinecap .08 W 1 0 1 RGB S\n");
// M(-1.,0)L(-10.1,0) fprintf(o,"0 setlinecap .08 W 1 0 1 RGB S\n");
// M(-1.6,0)L(-10.1,0) fprintf(o,"0 setlinecap .08 W 1 0 1 RGB S\n");

        M(-2.,0)L(-10.1,0) fprintf(o,"0 setlinecap .08 W 1 1 1 RGB S\n");

DO(m,15){M(-2.-.6*(m+.5),0)L(-2-.6*(m+1.),0)} fprintf(o,".09 W 1 0 1 RGB S\n");

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
        system("epstopdf e1etetma.eps");
        system( "open e1etetma.pdf"); //mac
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}
%\renewcommand\baselinestretch{.94}
%\renewcommand\baselinestretch{.94}
%\renewcommand\baselinestretch{1.15}
%\renewcommand\baselinestretch{1.2}
%\renewcommand\baselinestretch{1.2}

%\paperwidth150mm
%\paperwidth143mm
%\paperheight 257mm
%\paperheight 216mm
%\paperheight 106mm
%\paperheight 214mm
%\paperheight 106mm
%\textwidth 175mm
\paperwidth 404pt
\paperheight 300pt
%\oddsidemargin 5mm
%\oddsidemargin -72pt
\oddsidemargin -56pt
%\topmargin -17mm
%\topmargin -100pt
\topmargin -110pt
\pagestyle{empty}
\large
\usepackage{color}
\definecolor{rose}{rgb}{1,.9,1}
\renewcommand\thesection{\arabic{section}}
\renewcommand\figurename{Рис.}
%\renewcommand\refname{References}
%\renewcommand\figurename{Fig.}
\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 \bN {\mathbb{N}}
%\newcommand \bC {\mathbb{C}}
%\newcommand \bR {\mathbb{R}}
%\newcommand \cO {\mathcal{O}}
%\newcommand \cF {\mathcal{F}}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
%\newcommand \nS {\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!}
%\newcommand \pS {{~}~{~}}
%\newcommand \fac {\mathrm{Factorial}}
%\newcommand {\rf}[1] {(\ref{#1})}
%\newcommand{\iL}[1] {~\label{#1}\pS \rm[#1]\nS} %make the labels visible
%\newcommand{\iL}[1] {\label{#1}} %make the labels invisible; choose one of these options
%\newcommand \eL[1] {\iL{#1}\ee}
\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{e1etf}}
\put(0,0){\ing{e1etetma}}
 \figaxe
 \put( 4,225){\sx{1.3}{$u\!=\!3$}}
 \put( 22,288){\sx{1.1}{$u\!=\!2.8$}} \put(108,288){\sx{1.1}{$u\!=\!2.6$}}
 \put( 22, 8){\sx{1.1}{$u\!=\!2.8$}} \put(108, 8){\sx{1.1}{$u\!=\!2.6$}}
 \put(112,250){\sx{1.1}{$v\!=\!0.4$}}
 \put(120, 40){\sx{1.1}{$v\!=\!-0.4$}}
 %\put(140,230){\sx{1.1}{$p\!=\!2.4$}}
 \put(214,110){\sx{1.1}{$u\!=\!2.4$}}
 %\put(140, 64){\sx{1.1}{$p\!=\!2.4$}}
 %\put(160,198){\sx{1.1}{$q\!=\!0.2$}}
 \put(190,260){\sx{1.1}{$v\!=\!0.2$}}
 \put(130,156){\sx{1.3}{$u\!=\!2$}}
 \put(277, 148){\sx{1.3}{$v\!=\!0$}}
 \put(158,132){\sx{1.1}{$u\!=\!2.2$}}
 \put(190, 17){\sx{1.1}{$v\!=\!-0.2$}}
 \end{picture}}
 \end{document}
 \sx{.96}{\begin{picture}(410,320)
 %\put(0,0){\ing{e1egf} } % suppress this string to boost
 \put(0,0){\ing{e1esuma} } % suppress this string to boost
 \figaxe
 \put(100,194){\sx{1.4}{$u\!=\!3$}}
 \put(150,280){\sx{1.2}{$v\!=\!0.2$}}
 \put(150, 30){\sx{1.2}{$v\!=\!-0.2$}}
 \put(218,224){\sx{1.2}{$v\!=\!0.4$}}
 \put(259,266){\sx{1.2}{$u\!=\!2.8$}}%
 \put(258, 38){\sx{1.2}{$u\!=\!2.8$}}
 \put(306,260){\sx{1.2}{$u\!=\!2.6$}}
 \put(308, 39){\sx{1.2}{$u\!=\!2.6$}}
 \put(350,252){\sx{1.2}{$v\!=\!0.4$}}
% \put(390,239){\sx{1.2}{$u\!=\!2.4$}}
 \put(365,240){\sx{1.2}{$u\!=\!2.4$}}
% \put(370, 59){\sx{1.2}{$u\!=\!2.4$}}
% \put(390,205){\sx{1.2}{$v\!=\!0.2$}}
% \put(380, 88){\sx{1.2}{$v\!=\!-0.2$}}
 \put(352, 73){\sx{1.2}{$v\!=\!-0.4$}}
% \put(390,172){\sx{1.2}{$u\!=\!2.2$}}
% \put(390,122){\sx{1.2}{$u\!=\!2.2$}}
 \put(403,158){\sx{1.1}{$v\!=\!0$}}
 \put(403,143){\sx{1.1}{$v\!=\!0$}}
 \put(362,130){\sx{1.4}{$u\!=\!2$}}
 \put(174,165){\sx{1.2}{$u\!=\!3.2$}}
 \put( 60,148){\sx{1.4}{$v\!=\!0$}}
 \put(334,149){\sx{.9}{\bf huge values}}
 \end{picture}}

\end{document}

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