File:Ack4c.jpg

Complex map of tetration to Sheldon base $b\!=\! 1.52598338517+0.0178411853321 i$.

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

Another version of this image is loaded as http://mizugadro.mydns.jp/t/index.php/File:Tetsheldonmap03.png

C++ Generator of map
Files ado.cin, conto.cin, filog.cin, TetSheldonIma.inc, GLxw2048.inc should be loaded to the working directory in order to compile the code below. // using namespace std; typedef std::complex z_type;
 * 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.)
 * 4) include "conto.cin"
 * 5) include "filog.cin"

z_type b=z_type( 1.5259833851700000, 0.0178411853321000); z_type a=log(b); z_type Zo=Filog(a); z_type Zc=conj(Filog(conj(a))); DB A=32.;

z_type tetb(z_type z){ int k; DB t; z_type c, cu,cd; int K=2048; //#include "ima6.inc" z_type E[2048],G[2048]; DO(k,K){c=F[k]; E[k]=log(c)/a; G[k]=exp(a*c);} c=0.; z+=z_type(0.1196573712872846, 0.1299776198056910); DO(k,K){t=A*GLx[k];c+=GLw[k]*(G[k]/(z_type( 1.,t)-z)-E[k]/(z_type(-1.,t)-z));} cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) ); cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) ); c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd; return c;}
 * 1) include "GLxw2048.inc"
 * 1) include "TetSheldonIma.inc"

int main{ int j,k,m,m1,n; DB x,y, p,q, t; z_type z,c,d; //int M=161,M1=M+1; 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("tetsheldonmap.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); 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; 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("epstopdf tetsheldonmap.eps"); system( "open tetsheldonmap.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{$b=1.5 ~$} %\sx{.586} {\begin{picture}(620,216) \mapax \put(10,10){\ing{tet15ma}} %%%%%%%% \put(25,108.4){\sx{1.4}{\bf cut}}			\put(502,108.4){\sx{1.4}{$v\!=\!0$}} % \put(24,194){\sx{1.5}{$u\!+\!\mathrm i v \approx 2.306009391950\!+\! 1.081988656014\,\mathrm i$}} \put(24,19){\sx{1.5}{$u\!+\!\mathrm i v \approx 2.306009391950\!-\! 1.081988656014\,\mathrm i$}} %\put(20,16){\sx{1.5}{$u\!+\!\mathrm i v \approx 2.3\!-\!1.1\,\mathrm i$}} % \multiput(50,160)(143,10.4){4}{\sx{1.4}{$v\!=\!1$}}%% \multiput(48,129)(143,10.4){4}{\sx{1.4}{$v\!=\!0.8$}}%% \put(342,108){\sx{1.4}{$v\!=\!0$}}%% \multiput(46,56)(143,-10.4){4}{\sx{1.4}{$v\!=\!-1$}}%%

\put(341,96){\sx{1.4}{\rot{90}$u\!=\!2$\ero}} \put(381,96){\sx{1.4}{\rot{90}$u\!=\!3$\ero}} \end{picture}}%%%%%%%%% \end{document} \caption{$u\!+\!\mathrm i v\!=\!\mathrm{tet}_b(x\!+\!\mathrm i y)$ for $b\!=\!\sqrt{2} %\!\approx\! 1.41 $, $b\!=\!\exp(1/\mathrm e) %\!\approx\! 1.44 $, and $b\!=\!1.5~$ \label{maps1}} \end{figure} \end{document}

Refrences
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf 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://www.ils.uec.ac.jp/~dima/BOOK/202.pdf http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014. (In Russian)

D.Kouznetsov. Holomorphic ackermanns. 2015, in preparation.