Difference between revisions of "File:E1efig09abc1a150.png"

Latest revision as of 08:39, 13 December 2025

Summary

Figure 17.4 from page 245 of book «Superfunctions»^[1], 2020.

It appears also as page 251 of the Russian version «Суперфункции»^[2]

Complex maps of tetration \(\mathrm{tet}_b\) to base
\(b\!=\!1.5\) , left,
\(b\!=\!\exp(1/\mathrm e)\) , center, and
\(b\!=\!\sqrt{2}\) , right.

\(f\!=\! \mathrm{tet}(x\!+\!\mathrm i y)\) is shown in the \(x,y\) plane with levels \(~u\!=\!\Re(f)\!=\! \mathrm{const}~\) and levels \(~v\!=\!\Im(f)\!=\! \mathrm{const}~\). The integer levels are shown with thick lines.

This image is close to the figure 9 in the article ^[3]; the only notations at labels are improved.

Copyleft 2011,2020,2025 by Dmitrii Kouznetsov. You may copy, modify and/or distribute the image for free but the source should be attributed. Attribute your modifications if any.

C++ generator of first map, b=1.5

/* In order to compile the code below, the following files should be loaded: ado.cin, conto.cin, GLxw2048.inc, f2048b15.inc, f15.cin, plofu.cin */

 #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;

// #include <complex.h>
// #define z_type complex<double>
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)
 #include "conto.cin"
//DB T22=-8.5715740896774235522;
//DB T42= 9.6180745210214273558;
//#include "f21E.cin"
//#include "e1etf.cin"
#include "f15.cin"
 main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
// int M=201,M1=M+1;
// int N=401,N1=N+1;
 int M=81,M1=M+1;
 int N=161,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
// FILE *o;o=fopen("f21E.eps","w");  ado(o,0,0,94,92);
// FILE *o;o=fopen("02.eps","w");  ado(o,0,0,214,212);
// FILE *o;o=fopen("be1ezoom.eps","w");  ado(o,0,0,87,87);
 FILE *o;o=fopen("b15zoom.eps","w");  ado(o,87,87);
 fprintf(o,"46 45 translate\n 10 10 scale\n");
  DO(m,M1) X[m]=-4.+.1*(m-.5);
  DO(n,N1) Y[n]=-4.+.05*(n-.5);
 for(m=-4;m<5;m++) {M(m,-4)L(m,4)}
 for(n=-4;n<5;n++) {M(  -4,n)L(4,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
 fprintf(o,"/adobe-Roman findfont .6 scalefont setfont\n");
 for(m=-2;m<0;m+=2) {M(-4.7,m-.2) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<3;m+=2) {M(-4.4,m-.2) fprintf(o,"(%1d)s\n",m);}
 for(m=-2;m<0;m+=2) {M(m-.3,-4.48) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<3;m+=2) {M(m-.16,-4.48) fprintf(o,"(%1d)s\n",m);}
/*
 fprintf(o,"/Times-Italic findfont 1 scalefont setfont\n");
 //fprintf(o,"/adobe-italic findfont 1 scalefont setfont\n");
 M(-4.7,  4.5) fprintf(o,"(y)s\n");
 M(  4.6,-4.8) fprintf(o,"(x)s\n");
 M(-4,0)L(4.1,0) M(0,-4)L(0,4.1) fprintf(o,".01 W 1 0 1 RGB S\n");
*/
 
//z_type tm,tp,F[M1*N1];; 
 DO(m,M1)DO(n,N1){	g[m*N1+n]=9999;
 			f[m*N1+n]=9999;}
DB b=sqrt(2);
 DO(m,M1){x=X[m]; printf("x=%6.3f\n",x);
 DO(n,N1){y=Y[n]; z=z_type(x,y);
	if(abs(z+2.)>.04)
		{	
	//	c=F21E(z);
	//	c=E1ETF(z);
        	 c=F15(z);
 		p=Re(c); q=Im(c);
		if(p>-9999 && p<9999) g[m*N1+n]=p;
 		if(q>-9999 && q<9999 && fabs(q)>1.e-14) f[m*N1+n]=q;
		}
 	}}
fprintf(o,"1 setlinejoin 2 setlinecap\n");
p=2.; q=1.1;;
#include "plofu.cin"

fprintf(o,"0 setlinejoin 0 setlinecap\n");
// p=1.e-15;
// for(n=-10;n<11;n++) {q=p*n; z=z_type(q,0.); printf("%19.15f %19.15f\n",q,  Re(TQ2E(z))  )
; }
//y=2*M_PI/log(2.);
// y=M_PI/log(log(2));
//y=9.064720284;

//	M(-2,0)L(-10.1, 0) fprintf(o,"0.05 W 1 1 1 RGB S\n");
//
// DO(n,20){ M(-2.-.4*n,0)L(-2-.4*(n+.5),0) } 
//			fprintf(o,".11 W 0 0 0 RGB S\n");

//M(-2,0)L(-4.1, 0) fprintf(o,".1 W 0 0 0 RGB [.14 .14] 0 setdash S\n"); //fails at some pri
nters

//M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n");
 fprintf(o,"showpage\n");
 fprintf(o,"%cTrailer\n",'%');
 fclose(o);
//	system(      "gv b15zoom.eps &"); //for UNIX
	system(    "open b15zoom.eps"); //for macintosh
 	system("epstopdf b15zoom.eps"); 
// 	system(    "xpdf be1ezoom.pdf"); // for LINUX 
	getchar(); system("killall Preview"); // For macintosh
}

e1e09b.cc: C++ generator of second map

/* ado.cin, conto.cin, e1etf.cin should be loaded */

 #include <math.h>
 #include <stdio.h>
 #include <stdlib.h>
 #define DB double
 #define DO(x,y) for(x=0;x<y;x++)
#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 "conto.cin"
//DB T22=-8.5715740896774235522;
//DB T42= 9.6180745210214273558;
//#include "f21E.cin"
#include "e1etf.cin"
//#include "f15.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
// int M=201,M1=M+1;
// int N=401,N1=N+1;
 int M=81,M1=M+1;
 int N=161,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
// FILE *o;o=fopen("f21E.eps","w");  ado(o,0,0,94,92);
// FILE *o;o=fopen("02.eps","w");  ado(o,0,0,214,212);
// FILE *o;o=fopen("be1ezoom.eps","w");  ado(o,87,87);
 FILE *o;o=fopen("e1e09b.eps","w");  ado(o,87,87);
 fprintf(o,"46 45 translate\n 10 10 scale\n");
  DO(m,M1) X[m]=-4.+.1*(m-.5);
  DO(n,N1) Y[n]=-4.+.05*(n-.5);
 for(m=-4;m<5;m++) {M(m,-4)L(m,4)}
 for(n=-4;n<5;n++) {M(  -4,n)L(4,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
/*
 fprintf(o,"/adobe-Roman findfont .6 scalefont setfont\n");
 for(m=-2;m<0;m+=2) {M(-4.7,m-.2) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<3;m+=2) {M(-4.4,m-.2) fprintf(o,"(%1d)s\n",m);}
 for(m=-2;m<0;m+=2) {M(m-.3,-4.48) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<3;m+=2) {M(m-.16,-4.48) fprintf(o,"(%1d)s\n",m);}
*/
/*
 fprintf(o,"/Times-Italic findfont 1 scalefont setfont\n");
 //fprintf(o,"/adobe-italic findfont 1 scalefont setfont\n");
 M(-4.7,  4.5) fprintf(o,"(y)s\n");
 M(  4.6,-4.8) fprintf(o,"(x)s\n");
 M(-4,0)L(4.1,0) M(0,-4)L(0,4.1) fprintf(o,".01 W 1 0 1 RGB S\n");
*/
 
//z_type tm,tp,F[M1*N1];; 
 DO(m,M1)DO(n,N1){	g[m*N1+n]=9999;
 			f[m*N1+n]=9999;}
DB b=sqrt(2);
 DO(m,M1){x=X[m]; printf("x=%6.3f\n",x);
 DO(n,N1){y=Y[n]; z=z_type(x,y);
	if(abs(z+2.)>.04)
		{	
		//c=F21E(z);
		  c=E1ETF(z);
                // c=F15(z);
 		p=Re(c); q=Im(c);
		if(p>-9999 && p<9999) g[m*N1+n]=p;
 		if(q>-9999 && q<9999 && fabs(q)>1.e-14) f[m*N1+n]=q;
		}
 	}}
fprintf(o,"1 setlinejoin 2 setlinecap\n");
p=2.; q=1.1;;
#include "plofu.cin"

fprintf(o,"0 setlinejoin 0 setlinecap\n");
// p=1.e-15;
// for(n=-10;n<11;n++) {q=p*n; z=z_type(q,0.); printf("%19.15f %19.15f\n",q,  Re(TQ2E(z))  ); }
//y=2*M_PI/log(2.);
// y=M_PI/log(log(2));
//y=9.064720284;

//	M(-2,0)L(-10.1, 0) fprintf(o,"0.05 W 1 1 1 RGB S\n");
//
// DO(n,20){ M(-2.-.4*n,0)L(-2-.4*(n+.5),0) } 
//			fprintf(o,".11 W 0 0 0 RGB S\n");

//M(-2,0)L(-4.1, 0) fprintf(o,".1 W 0 0 0 RGB [.14 .14] 0 setdash S\n"); //fails at some printers

//M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n");
 fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
 	system("epstopdf e1e09b.eps"); 
	system(    "open e1e09b.pdf");
	getchar(); system("killall Preview"); // For macintosh
}

C++ generator of third map, b=sqrt(2) approx 1.41

/* Files required: ado.cin, conto.cin, sqrt2f21e.cin */

 #include <math.h>
 #include <stdio.h>
 #include <stdlib.h>
 #define DB double
 #define DO(x,y) for(x=0;x<y;x++)
#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 "conto.cin"
DB T22=-8.5715740896774235522;
DB T42= 9.6180745210214273558;
//#include "f21E.cin"
#include "sqrt2f21e.cin"
//#include "e1etf.cin"
//#include "f15.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
// int M=201,M1=M+1;
// int N=401,N1=N+1;
 int M=81,M1=M+1;
 int N=161,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
// FILE *o;o=fopen("f21E.eps","w");  ado(o,0,0,94,92);
// FILE *o;o=fopen("02.eps","w");  ado(o,0,0,214,212);
// FILE *o;o=fopen("bq2zoom.eps","w");  ado(o,87,87);
 FILE *o;o=fopen("e1e09q2.eps","w");  ado(o,87,87);
 fprintf(o,"46 45 translate\n 10 10 scale\n");
  DO(m,M1) X[m]=-4.+.1*(m-.5);
  DO(n,N1) Y[n]=-4.+.05*(n-.5);
 for(m=-4;m<5;m++) {M(m,-4)L(m,4)}
 for(n=-4;n<5;n++) {M(  -4,n)L(4,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
/*
 fprintf(o,"/adobe-Roman findfont .6 scalefont setfont\n");
 for(m=-2;m<0;m+=2) {M(-4.7,m-.2) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<3;m+=2) {M(-4.4,m-.2) fprintf(o,"(%1d)s\n",m);}
 for(m=-2;m<0;m+=2) {M(m-.3,-4.48) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<3;m+=2) {M(m-.16,-4.48) fprintf(o,"(%1d)s\n",m);}
*/
/*
 fprintf(o,"/Times-Italic findfont 1 scalefont setfont\n");
 //fprintf(o,"/adobe-italic findfont 1 scalefont setfont\n");
 M(-4.7,  4.5) fprintf(o,"(y)s\n");
 M(  4.6,-4.8) fprintf(o,"(x)s\n");
 M(-4,0)L(4.1,0) M(0,-4)L(0,4.1) fprintf(o,".01 W 1 0 1 RGB S\n");
*/
 
//z_type tm,tp,F[M1*N1];; 
 DO(m,M1)DO(n,N1){	g[m*N1+n]=9999;
 			f[m*N1+n]=9999;}
DB b=sqrt(2);
 DO(m,M1){x=X[m]; printf("x=%6.3f\n",x);
 DO(n,N1){y=Y[n]; z=z_type(x,y);
	if(abs(z+2.)>.04)
		{	
		c=F21E(z);
		//c=E1ETF(z);
                // c=F15(z);
 		p=Re(c); q=Im(c);
		if(p>-9999 && p<9999) g[m*N1+n]=p;
 		if(q>-9999 && q<9999 && fabs(q)>1.e-14) f[m*N1+n]=q;
		}
 	}}
fprintf(o,"1 setlinejoin 2 setlinecap\n");
p=2.; q=1.1;;
#include "plofu.cin"

fprintf(o,"0 setlinejoin 0 setlinecap\n");
// p=1.e-15;
// for(n=-10;n<11;n++) {q=p*n; z=z_type(q,0.); printf("%19.15f %19.15f\n",q,  Re(TQ2E(z))  ); }
//y=2*M_PI/log(2.);
// y=M_PI/log(log(2));
//y=9.064720284;

//	M(-2,0)L(-10.1, 0) fprintf(o,"0.05 W 1 1 1 RGB S\n");
//
// DO(n,20){ M(-2.-.4*n,0)L(-2-.4*(n+.5),0) } 
//			fprintf(o,".11 W 0 0 0 RGB S\n");

//M(-2,0)L(-4.1, 0) fprintf(o,".1 W 0 0 0 RGB [.14 .14] 0 setdash S\n"); //fails at some printers

//M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n");
 fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
 	system("epstopdf e1e09q2.eps"); 
 	system(    "open e1e09q2.pdf");
	getchar(); system("killall Preview"); // For macintosh
}

Latex generator of labels

\documentclass[12pt,oneside]{book}
%\documentclass{standalone}
%\usepackage{cmap}
\usepackage{geometry}
\paperheight 520pt
\paperwidth 1520pt
%\paperwidth  1600pt
\textwidth 3800pt
\topmargin -92pt
\oddsidemargin -86pt
%\usepackage[utf8]{inputenc}
%\usepackage[T2A]{fontenc}
%\usepackage[english]{babel}
%\usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx}
\usepackage{rotating}
\large

\usepackage{color}
\newcommand \sx {\scalebox}
\newcommand \rme {{\rm e}}	%makes the base of natural logarithms Roman font 
\newcommand \rmi {{\rm i}}	%imaginary unity is always roman font
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing \includegraphics
\begin{document}

%\sx{1.65}{\begin{picture}(82,84) 
\sx{6}{\begin{picture}(82,84) 
\put(0,0){\ing{e1e09a15}}
\put(2,82){\sx{.5}{$y$}}
\put(2,73){\sx{.5}{$3$}}
\put(2,63){\sx{.5}{$2$}}
\put(2,53){\sx{.5}{$1$}}
\put(2,43){\sx{.5}{$0$}}
\put(-2,33){\sx{.5}{$-1$}}
\put(-2,23){\sx{.5}{$-2$}}
\put(-2,13){\sx{.5}{$-3$}}
\put(-2, 3){\sx{.5}{$-4$}}
\put( 1,0){\sx{.5}{$-4$}}
\put(11,0){\sx{.5}{$-3$}}
\put(21,0){\sx{.5}{$-2$}}
\put(31,0){\sx{.5}{$-1$}}
\put(45,0){\sx{.5}{$0$}}
\put(55,0){\sx{.5}{$1$}}
\put(65,0){\sx{.5}{$2$}}
\put(75,0){\sx{.5}{$3$}}
\put(84,.4){\sx{.5}{$x$}} 
\put(48,74){\sx{.5}{\rot{71}$v\!=\!1$\ero}}
\put(56,72){\sx{.5}{\rot{39}$v\!=\!0.8$\ero}}
\put(62,65.6){\sx{.5}{\rot{23}$v\!=\!0.6$\ero}}
\put(65,58){\sx{.5}{\rot{13}$v\!=\!0.4$\ero}}
\put(66,50.6){\sx{.5}{\rot{6}$v\!=\!0.2$\ero}}
\put(66,43.6){\sx{.5}{$v\!=\!0$}}
\put(64,37){\sx{.5}{\rot{-7}$v\!=\!-0.2$\ero}}
\put(63,30){\sx{.5}{\rot{-12}$v\!=\!-0.4$\ero}}
\put(43.4,21){\sx{.5}{\rot{-70}$v\!=\!-1$\ero}}
\put(43,32){\sx{.5}{\rot{63}$u\!=\!1$\ero}}
\put(46,26){\sx{.5}{\rot{53}$u\!=\!1.4$\ero}}
\put(49,21){\sx{.5}{\rot{50}$u\!=\!1.6$\ero}}
\put(52,14.4){\sx{.5}{\rot{47}$u\!=\!1.8$\ero}}
\put(58,6){\sx{.5}{\rot{46}$u\!=\!2$\ero}}
 \end{picture}}
\sx{6}{\begin{picture}(82,84) 
%\sx{1.65}{\begin{picture}(82,84) 
\put(0,0){\ing{e1e09b}} 
\put(11,0){\sx{.5}{$-3$}}
\put(21,0){\sx{.5}{$-2$}}
\put(31,0){\sx{.5}{$-1$}}
\put(45,0){\sx{.5}{$0$}}
\put(55,0){\sx{.5}{$1$}}
\put(65,0){\sx{.5}{$2$}}
\put(75,0){\sx{.5}{$3$}}
\put(84,.4){\sx{.5}{$x$}} 
\put(42,60){\sx{.5}{\rot{86}$v\!=\!1$\ero}}
\put(46,61){\sx{.5}{\rot{79}$v\!=\!0.8$\ero}}
\put(52,62){\sx{.5}{\rot{64}$v\!=\!0.6$\ero}}
\put(58,59){\sx{.5}{\rot{46}$v\!=\!0.4$\ero}}
\put(63,52){\sx{.5}{\rot{26}$v\!=\!0.2$\ero}}
\put(66,43.6){\sx{.5}{$v\!=\!0$}}
\put(60,36){\sx{.5}{\rot{-27}$v\!=\!-0.2$\ero}}
\put(42,14){\sx{.5}{\rot{90}$v\!=\!-1$\ero}}
\put(42,32.3){\sx{.5}{\rot{56}$u\!=\!1$\ero}}
\put(44,26){\sx{.5}{\rot{42}$u\!=\!1.4$\ero}}
\put(47,22){\sx{.5}{\rot{37}$u\!=\!1.6$\ero}}
\put(53,17){\sx{.5}{\rot{32}$u\!=\!1.8$\ero}}
\put(59,9){\sx{.5}{\rot{18}$u\!=\!2$\ero}}
\end{picture}}
\sx{6}{\begin{picture}(80,84) 
\put(0,0){\ing{e1e09q2}} 
\put(11,0){\sx{.5}{$-3$}}
\put(21,0){\sx{.5}{$-2$}}
\put(31,0){\sx{.5}{$-1$}}
\put(45,0){\sx{.5}{$0$}}
\put(55,0){\sx{.5}{$1$}}
\put(65,0){\sx{.5}{$2$}}
\put(75,0){\sx{.5}{$3$}}
\put(84,.4){\sx{.5}{$x$}} 
\put(40.6,57){\sx{.5}{\rot{86}$v\!=\!1$\ero}}
\put(44,58){\sx{.5}{\rot{80}$v\!=\!0.8$\ero}}
\put(49,61){\sx{.5}{\rot{75}$v\!=\!0.6$\ero}}
\put(57,61){\sx{.5}{\rot{70}$v\!=\!0.4$\ero}}
\put(63,54){\sx{.5}{\rot{42}$v\!=\!0.2$\ero}}
\put(66,43.6){\sx{.5}{$v\!=\!0$}}
\put(66,32){\sx{.5}{\rot{-60}$v\!=\!-0.2$\ero}}
\put(24,4.8){\sx{.5}{\rot{31}$v\!=\!-1$\ero}}
\put(42,32.3){\sx{.5}{\rot{56}$u\!=\!1$\ero}}
\put(47,28){\sx{.5}{\rot{48}$u\!=\!1.4$\ero}}
\put(49,24){\sx{.5}{\rot{33}$u\!=\!1.6$\ero}}
\put(50,17){\sx{.5}{\rot{13}$u\!=\!1.8$\ero}}
\put(50,9){\sx{.5}{\rot{-13}$u\!=\!2$\ero}}
\end{picture}}
\end{document}

References

↑ https://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862
https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3
https://mizugadro.mydns.jp/BOOK/458.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Piblishing, 2020.
↑ https://mizugadro.mydns.jp/BOOK/202.pdf Дмитрий Кузнецов. Суперфункции. Lambert Academic Piblishing, 2014.
↑ https://www.ams.org/journals/mcom/2012-81-280/S0025-5718-2012-02590-7/S0025-5718-2012-02590-7.pdf
https://www.researchgate.net/publication/51892955_Computation_of_the_Two_Regular_Super-Exponentials_to_base_exp1e
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, 2012, 81, February 8. p.2207-2227.

Keywords

«Superfunction», «Superfunctions», «Tetration»,

File history

Click on a date/time to view the file as it appeared at that time.

	Date/Time	Thumbnail	Dimensions	User	Comment
current	08:37, 13 December 2025		1,514 × 518 (364 KB)	T (talk \| contribs)	more labels
	17:50, 20 June 2013		2,234 × 711 (883 KB)	Maintenance script (talk \| contribs)	Importing image file

You cannot overwrite this file.

File usage

The following page uses this file:

Tetration

[1] ttps://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862
https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3
https://mizugadro.mydns.jp/BOOK/458.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Piblishing, 2020.

[2] ttps://mizugadro.mydns.jp/BOOK/202.pdf Дмитрий Кузнецов. Суперфункции. Lambert Academic Piblishing, 2014.

[e1e-3] ttps://www.ams.org/journals/mcom/2012-81-280/S0025-5718-2012-02590-7/S0025-5718-2012-02590-7.pdf
https://www.researchgate.net/publication/51892955_Computation_of_the_Two_Regular_Super-Exponentials_to_base_exp1e
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, 2012, 81, February 8. p.2207-2227.

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