File:Ack3c600.jpg

Summary

Title / Description	Complex map of tetration to base ${\displaystyle b=1.5}$ ${\displaystyle u+\mathrm {i} v=\mathrm {tet} _{b}(x\!+\!\mathrm {i} y)}$
Citizendium author & Copyright holder	Copyright © Dmitrii Kouznetsov. See below for licence/re-use information.
Date created	August 2014
Country of first publication	Japan
Notes	I plan to use this image in the article D.Kouznetsov. Holomorphic ackermann. 2015, in preparation.
Other versions	http://mizugadro.mydns.jp/t/index.php/File:Ack3a600.jpg The real-real plot of this function is one of curves at http://en.citizendium.org/wiki/File:Tetreal10bx10d.png
Using this image on CZ	Please click here to add the credit line, then copy the code below to add this image to a Citizendium article, changing the size, alignment, and caption as necessary. {{Image|Ack3c600.jpg|right|350px|Add image caption here.}}

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C++ Generator of map]

Files ado.cin, conto.cin, f15.cin, f2048b15.inc, GLxw2048.inc should be loaded to the working directory in order to compile the code below.

#include <math.h> #include <stdio.h> #include <stdlib.h> #define DB double #define DO(x,y) for(x=0;x<y;x++) #include <complex> #define z_type std::complex<double> #define Re(x) x.real() #define Im(x) x.imag() #define I z_type(0.,1.) #include "f15.cin" #include "conto.cin" int main(){ int j,k,m,m1,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; 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("tet15ma.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<N1;n++) Y[n]=-10.+.05*(n-1.); for(m=-30;m<31;m++){if(m==0){M(m,-10.2)L(m,10.2)} else{M(m,-10)L(m,10)}} for(n=-10;n<11;n++){ M( -30,n)L(30,n)} fprintf(o,".008 W 0 0 0 RGB S\n"); DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;} DO(n,N1){y=Y[n]; for(m=295;m<305;m++) {x=X[m]; //printf("%5.2f\n",x); z=z_type(x,y); c=F15(z); p=Re(c);q=Im(c); if(p>-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(d*log(1.5)); 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)/log(1.5); 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"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf tet15ma.eps"); system( "open tet15ma.pdf"); getchar(); system("killall Preview"); }

Latex Generator of labels]

\documentclass{amsproc} \usepackage{graphicx} \usepackage{rotating} \usepackage{hyperref} \newcommand \be {\begin{eqnarray}} \newcommand \ee {\end{eqnarray} } \newcommand \sx {\scalebox} \newcommand \rme {{\rm e}} \newcommand \rmi {{\rm i}} \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{\iL}[1] {~\label{#1}\pS \rm[#1]\nS} \newcommand \eL[1] {\iL{#1}\ee} \newcommand \ing \includegraphics \newcommand \tet {\mathrm{tet}} \usepackage{geometry} \topmargin -97pt \oddsidemargin -87pt \paperwidth 618pt \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}{$x$}} } {\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}{$u\!+\!\mathrm i v \approx 2.306009391950\!-\! 1.081988656014\,\mathrm i$}} \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}

References