File:Ack3a600.jpg

Summary

Title / Description	Complex map of tetration to base ${\displaystyle b={\sqrt {2}}}$ ${\displaystyle u+\mathrm {i} v=\mathrm {tet} _{b}(x\!+\!\mathrm {i} y)\$}$
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|Ack3a600.jpg|right|350px|Add image caption here.}}

Licensing

   

This media, Ack3a600.jpg, is licenced under the Creative Commons Attribution 3.0 Unported License

You are free: To Share — To copy, distribute and transmit the work; To Remix — To adapt the work.
Under the following conditions: Attribution — You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).
For any reuse or distribution, you must make clear to others the licence terms of this work (the best way to do this is with a link to this licence's web page). Any of the above conditions can be waived if you get permission from the copyright holder. Nothing in this licence impairs or restricts the author's moral rights.
Read the full licence.

C++ generator of map

Files ado.cin, conto.cin, sqrt2f21e.cin 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 "sqrt2f21e.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]; char v[M1*N1]; // v is working array FILE *o;o=fopen("tetqma.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=F21E(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*(.5*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)*(2./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"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf tetqma.eps"); system( "open tetqma.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 \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 \eL[1] {\iL{#1}\ee} \newcommand \ing \includegraphics \newcommand \tet {\mathrm{tet}} \usepackage{geometry} \topmargin -94pt \oddsidemargin -87pt \paperwidth 618pt \paperheight 216pt \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}{$x$}} } {\begin{picture}(620,216) \mapax \put(10,10){\ing{tetqma}} \put(22,194){\sx{1.4}{$v\!=\!0$}} \put(254,206){\sx{1.4}{$v\!=\!-0.2$}} \put(262,182){\sx{1.4}{$v\!=\!0.2$}} \put(262,170){\sx{1.4}{$v\!=\!0.4$}} \put(266,148){\sx{1.4}{$v\!=\!1$}} \put(261,69){\sx{1.4}{$v\!=\!-1$}} \put(308,158){\sx{1.4}{\rot{66}$u\!=\!2.4$\ero}} \put(324,158){\sx{1.4}{\rot{53}$u\!=\!2.2$\ero}} \put(314,138){\sx{1.4}{\rot{-33}$u\!=\!1.8$\ero}} \put(318,61){\sx{1.4}{\rot{-56}$u\!=\!2.2$\ero}} \put(214,12){\sx{1.4}{\rot{80}$u\!=\!3.8$\ero}} \put(240,12){\sx{1.4}{\rot{83}$u\!=\!3.6$\ero}} \put(256,12){\sx{1.4}{\rot{85}$u\!=\!3.4$\ero}} \put(282,16){\sx{1.4}{\rot{90}$u\!=\!3$\ero}} \put(294,12){\sx{1.4}{\rot{90}$u\!=\!2.8$\ero}} \put(22, 145.6){\sx{1.4}{$u\!=\!4$}} \put(502, 151){\sx{1.4}{$u\!=\!2$}} \put(25,108.4){\sx{1.4}{\bf cut}} \put(502,108.4){\sx{1.4}{$v\!=\!0$}} \put(22, 70.4){\sx{1.4}{$u\!=\!4$}} \put(502, 64.6){\sx{1.4}{$u\!=\!2$}} \put(22, 21){\sx{1.4}{$v\!=\!0$}} \end{picture}} \end{document}

References

http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html
http://www.ils.uec.ac.jp/~dima/PAPERS/2010q2.pdf
http://mizugadro.mydns.jp/PAPERS/2010q2.pdf D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.

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.