File:Ack4a600.jpg

Complex map of tetration to base $b\!=\!2$

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

C++ Generator of map
Files ado.cin, conto.cin, tet2f4c.cin, Tet2f2048.inc, GLxw2048.inc should be loaded to the working directory in order to compile the code below.
 * 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.)
 * 11) include "tet2f4c.cin"
 * 12) 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); 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 tet2ma.eps"); system( "open tet2ma.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 \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 \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing \includegraphics \usepackage{geometry} %\topmargin -94pt \topmargin -97pt \oddsidemargin -87pt \paperwidth 618pt %\paperheight 216pt \paperheight 214pt

\begin{document}

\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}{$x$}} } %\sx{.586} {\begin{picture}(620,216) \put(10,10){\ing{tet2ma}} \mapax \multiput(110,118)(56.1,10.7){8}{\sx{1.2}{$u\!=\!0.8$}} \multiput(256,120)(56.1,10.7){7}{\sx{1.2}{\rot{20}$u\!=\!1$\ero}} \multiput(302,120)(56.1,10.7){6}{\sx{1.2}{\rot{0}$v\!=\!1$\ero}} \put(25,108.4){\sx{1.4}{\bf cut}}		\put(302,108.4){\sx{1.2}{$v\!=\!0$}} \multiput(124,92)(56.1,-10.7){7}{\sx{1.2}{$v\!=\!-1.6$}}

\put(20,200){\sx{1.4}{$u+\mathrm i v \approx 0.8246785461+ 1.5674321238 \,\mathrm i$}} \put(30, 20){\sx{1.4}{$u+\mathrm i v \approx 0.8246785461 - 1.5674321238 \,\mathrm i$}} \end{picture}} \end{document}

Refrences
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html D.Kouznetsov. (2009). Solutions of F(z+1)=exp(F(z)) in the complex plane. Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7.

https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфунцкии. Lambert Academic Publishing, 2014. (In Russian)