File:Ack4bFragment.jpg

Fragment of image http://mizugadro.mydns.jp/t/index.php/File:Ack4b600.jpg

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

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

Usage
This is fragment of image fig.3b (with improved resolution) of publication "Evaluation of holomorphic ackermanns", 2014.

C++ Generator of map]
Files ado.cin, conto.cin, fsexp.cin 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.)

//#include "tet2f4c.cin"
 * 1) include "conto.cin"


 * 1) include "filog.cin"

//z_type b=z_type( 1.5259833851700000, 0.0178411853321000); z_type b=M_E; /* z_type a=log(b); z_type Zo=Filog(a); z_type Zc=conj(Filog(conj(a))); DB A=32.; //#include "tet2f4c.cin"
 * 1) include "fsexp.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=641,M1=M+1; int N=402,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("tet2m2.eps","w");ado(o,1604,804); // FILE *o;o=fopen("tettenm2.eps","w");ado(o,1604,804); // FILE *o;o=fopen("amsfig4dFragmen.eps","w");ado(o,1604,804); // FILE *o;o=fopen("amsfig4aFragmen.eps","w");ado(o,1604,804); FILE *o;o=fopen("amsfig4bFragmen.eps","w");ado(o,1604,804); fprintf(o,"802 402 translate\n 100 100 scale 2 setlinecap 1 setlinejoin\n"); DO(m,M1)X[m]=-8.+.05*(m-.3); DO(n,200)Y[n]=-4.+.02*n; Y[200]=-.01; Y[201]= .01; for(n=202;n-99999. && p<99999. && q>-99999. && q<99999. ){ g[m*N1+n]=p;f[m*N1+n]=q;} d=c; for(k=1;k<31;k++) { m1=m+k*20; if(m1>M) break; //               d=exp(a*d); //               d=exp(d*log(2.)); d=exp(d); p=Re(d);q=Im(d); if(p>-99999. && p<99999. && q>-99999. && q<99999. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;} }          d=c; for(k=1;k<31;k++) { m1=m-k*20; if(m1<0) break; d=log(d); //             d=log(d)/a; //               d=log(d)/log(2.); p=Re(d);q=Im(d); if(p>-99999. && p<99999. && q>-99999. && q<99999. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;} }       }} fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=20;q=1; for(m=-4;m<4;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".004 W 0 .6 0 RGB S\n"); for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".004 W .9 0 0 RGB S\n"); for(m=0;m<4;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".004 W 0 0 .9 RGB S\n"); for(m=1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".03 W .9 0 0 RGB S\n"); for(m=1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".03 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".03 W .6 0 .6 RGB S\n"); for(m=-4;m<5;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".03 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 amsfig4bFragmen.eps"); system( "open amsfig4bFragmen.pdf"); getchar; system("killall Preview"); }

Latex Generator of labels]
\documentclass{amsproc} \usepackage{graphicx} \usepackage{rotating} \usepackage{hyperref} \newcommand \sx {\scalebox} \newcommand \rme 	 %% %\newcommand \rme 	%% \newcommand \rmi 	 %%imaginary unity \newcommand \ds {\displaystyle} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing \includegraphics \usepackage{geometry} \topmargin -94pt \oddsidemargin -70pt \paperwidth 1666pt \paperheight 856pt \textwidth 1900px \textheight 900px \begin{document} \parindent 0pt

\newcommand \mapax { \put(18,820){\sx{5}{$y$}} \put(18,730){\sx{5}{$3$}} \put(18,630){\sx{5}{$2$}} \put(18,530){\sx{5}{$1$}} \put(18,430){\sx{5}{$0$}} \put(-14,329){\sx{5}{$-1$}} \put(-14,229){\sx{5}{$-2$}} \put(-14,129){\sx{5}{$-3$}} \put(-14, 29){\sx{5}{$-4$}} \put(14, 0){\sx{5}{$-8$}} \put(114, 0){\sx{5}{$-7$}} \put(214, 0){\sx{5}{$-6$}} \put(314, 0){\sx{5}{$-5$}} \put(414, 0){\sx{5}{$-4$}} \put(514, 0){\sx{5}{$-3$}} \put(614, 0){\sx{5}{$-2$}} \put(714, 0){\sx{5}{$-1$}} \put(844, 0){\sx{5}{$0$}} \put(944, 0){\sx{5}{$1$}} \put(1044, 0){\sx{5}{$2$}} \put(1144, 0){\sx{5}{$3$}} \put(1244, 0){\sx{5}{$4$}} \put(1344, 0){\sx{5}{$5$}} \put(1444, 0){\sx{5}{$6$}} \put(1544, 0){\sx{5}{$7$}} \put(1634, 0){\sx{5}{$x$}} } %\flushright{$b=\mathrm e \approx 2.71$}

{\begin{picture}(1620,850) %%% \put(50,40){\ing{amsfig4bFragmen}}  \mapax \put(114,660){\sx{8}{$b=\mathrm e$}} \put(76,798){\sx{4}{$u\!+\!\mathrm i v \approx 0.318131505204764\!+\! 1.33723570143069 \,\mathrm i$}} \put(80,90){\sx{4}{$u\!+\!\mathrm i v \approx 0.318131505204764 \!-\! 1.33723570143069 \,\mathrm i$}} \put(60,434){\sx{4}{\bf cut}} \put(760,434){\sx{4}{$v\!=\!0$}} \multiput(46,550)(448,105){3}{\sx{4}{$v\!=\!1.4$}} \multiput(268,584)(448,105){3}{\sx{4}{$u\!=\!0.4$}} \multiput(46,316)(448,-105){4}{\sx{4}{$v\!=\!-1.4$}} \multiput(298,464)(448,105){4}{\sx{4}{$v\!=\!1$}} \multiput(290,404)(448,-105){4}{\sx{4}{$v\!=\!-1$}}

\end{picture}} \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). Solution of F(z+1)=exp(F(z)) in the complex z-plane. Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7.

http://www.ils.uec.ac.jp/~dima/PAPERS/2010vladie.pdf http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.

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)

http://mizugadro.mydns.jp/BOOK/437.pdf D.Kouznetsov. Suparfuncctions. Mizugadro, 2015. (In English)

http://myweb.astate.edu/wpaulsen/tetration.html William Paulsen. Tetration is repeated exponentiation. (2016). We can define $^0b = 1, ^1b = b, ^2b = b^b$, &3b = b^{b^b}$, etc. ..