File:E1etetma8.jpg

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)

and in the article

C++ generator of map
//using namespace std; typedef std::complex z_type; //#include "e1egf.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; } }}
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 1) include "e1etf.cin"
 * 2) include "conto.cin"

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]   	% make the English version \newcommand \EN[1] –     	% suppress the English version \newcommand \RU[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 	 %%makes the base of natural logarithms Roman font %\newcommand \rme 	%%makes the base of natural logarithms Italics font; choose one of these \newcommand \rmi 	 %%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}