File:Ack3b600.jpg

Complex map of tetration to base $b\!=\!\exp(1/\mathrm e)\!\approx\!1.44$

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

C++ Generator of map]
Files ado.cin, conto.cin, e1etf.cin should be loaded to the working directory in order to compile the code below. //#include "tet2f4c.cin"
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 7) define z_type std::complex
 * 8) define Re(x) x.real
 * 9) define Im(x) x.imag
 * 10) define I z_type(0.,1.)
 * 1) include "e1etf.cin"
 * 2) include "conto.cin"

int main{ int j,k,m,m1,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; //z_type Zo=z_type(.31813150520476413, 1.3372357014306895); //z_type Zc=z_type(.31813150520476413,-1.3372357014306895);

int M=601,M1=M+1; int N=461,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("20.eps","w");ado(o,602,202); // FILE *o;o=fopen("tet2ma.eps","w");ado(o,602,202); FILE *o;o=fopen("tet1ma.eps","w");ado(o,602,202); fprintf(o,"301 101 translate\n 10 10 scale\n"); DO(m,M1)X[m]=-30.+.1*(m); DO(n,200)Y[n]=-10.+.05*n; Y[200]=-.01; Y[201]= .01; for(n=202;n-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;} /*          d=c; for(k=1;k<31;k++) { m1=m+k*10; if(m1>M) break; //               d=exp(a*d); d=exp(d*log(2.)); p=Re(d);q=Im(d); if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;} }          d=c; for(k=1;k<31;k++) { m1=m-k*10; if(m1<0) break; //             d=log(d)/a; d=log(d)/log(2.); p=Re(d);q=Im(d); if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;} }       }}

fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1;q=.5; for(m=-10;m<10;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<10;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<10;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<10;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".08 W .9 0 0 RGB S\n"); for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".08 W .6 0 .6 RGB S\n"); for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 0 RGB S\n"); // y= 0; for(m=0;m<260;m+=6) {x=-2.-.1*m; M(x,y) L(x-.1,y)} // fprintf(o,".07 W 1 .5 0 RGB S\n"); // y= 0; for(m=3;m<260;m+=6) {x=-2-.1*m; M(x,y) L(x-.1,y)} // fprintf(o,".07 W 0 .5 1 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); //system( "ggv fig3.eps"); system("epstopdf tet1ma.eps"); system( "open tet1ma.pdf"); getchar; system("killall Preview"); }

Latex Generator of labels]
\documentclass{amsproc} \usepackage{graphicx}	% Use pdf, png, jpg, or eps§ with pdflatex; use eps in DVI mode\usepackage{amssymb} \usepackage{rotating} \usepackage{hyperref} \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}}

\usepackage{geometry} %\topmargin -94pt \topmargin -97pt \oddsidemargin -87pt \paperwidth 618pt %\paperheight 216pt \paperheight 214pt

\begin{document} %\title{AMS Proceedings Series Sample}

\newcommand \mapax { \put(2,206){\sx{1.2}{$y$}} \put(2,188){\sx{1.2}{$8$}} \put(2,168){\sx{1.2}{$6$}} \put(2,148){\sx{1.2}{$4$}} \put(2,128){\sx{1.2}{$2$}} \put(2,108){\sx{1.2}{$0$}} \put(-6,88){\sx{1.2}{$-2$}} \put(-6,68){\sx{1.2}{$-4$}} \put(-6,48){\sx{1.2}{$-6$}} \put(-6,28){\sx{1.2}{$-8$}} \put(-1,1){\sx{1.2}{$-30$}} \put( 49,1){\sx{1.2}{$-25$}} \put( 99,1){\sx{1.2}{$-20$}} \put(149,1){\sx{1.2}{$-15$}} \put(199,1){\sx{1.2}{$-10$}} \put(252,1){\sx{1.2}{$-5$}} \put(309,1){\sx{1.2}{$0$}} \put(329,1){\sx{1.2}{$2$}} \put(349,1){\sx{1.2}{$4$}} \put(369,1){\sx{1.2}{$6$}} \put(389,1){\sx{1.2}{$8$}} \put(407,1){\sx{1.2}{$10$}} \put(457,1){\sx{1.2}{$15$}} \put(507,1){\sx{1.2}{$20$}} \put(557,1){\sx{1.2}{$25$}} \put(607,1){\sx{1.2}{$x$}} }

%\flushright{$b=\exp(1/\mathrm e)\approx 1.44$} %\sx{.586} {\begin{picture}(620,216) \mapax \put(10,10){\ing{tet1ma}} %%%%%%%% \put(25,108.4){\sx{1.4}{\bf cut}}			\put(502,108.4){\sx{1.4}{$v\!=\!0$}} %%%%%%%%%%%% \put(262,192){\sx{1.4}{$v\!=\!0.2$}} \put(260,170){\sx{1.4}{$v\!=\!0.4$}} \put(266,156){\sx{1.4}{$v\!=\!1$}} \put(263,60){\sx{1.4}{$v\!=\!-1$}} % \put(100,156){\sx{1.4}{\rot{45}$u\!=\!3$\ero}} \put(102,55){\sx{1.4}{\rot{-44}$u\!=\!3$\ero}} % \put(356,96){\sx{1.4}{\rot{90}$u\!=\!2$\ero}} \put(385,96){\sx{1.4}{\rot{90}$u\!=\!2.2$\ero}} \put(449,96){\sx{1.4}{\rot{90}$u\!=\!2.4$\ero}} % \put(375,146){\sx{1.4}{\rot{51}$v\!=\!0.2$\ero}} \put(367,76){\sx{1.4}{\rot{-56}$v\!=\!-0.2$\ero}} \end{picture}} \end{document}

Refrences
http://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2012-02590-7/S0025-5718-2012-02590-7.pdf 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, 2012 February 8. ISSN 1088-6842(e) ISSN 0025-5718(p)

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 Publishing, 2014. (In Russian)