File:Sqrt2atemap.jpg

Fig.16.3 from page 221 of book «Superfunctions» ^[1], 2020.

This image appears also as Рис.16.3 of the Russian version Суперфункции ^[2], 2014.

This image is used also in Рис.2 of article ^[3] at Mathematics of Computation, 2010.

The picture shows the Complex map of arctetration to base \(\sqrt{2}\):

\(u\!+\!\mathrm i v = \mathrm{ate}_{\sqrt{2}}(x\!+\!\mathrm i y)\)

The arctetration ate is inverse function of tetration tet to the same base.

The Arctetration satisfies the Abel equation

\( \mathrm{ate}_{\sqrt{2}}(\exp_{\sqrt{2}}(z))=\mathrm{ate}_{\sqrt{2}}(z)+1 \)

In this equation, \( \exp_{\sqrt{2}} \) appears as transfer function; then, \( \mathrm{ate}_{\sqrt{2}} \) is its Abel function.

The special case «Base sqrt2» is interesting because for this base, the both the fixed point of the logarithm are integer.

C++ generator of the First map

/* Files ado.cin, conto.cin, sqrt2f21e.cin should be loaded 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>
 typedef std::complex<double> z_type;
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)
 #include "conto.cin"
// #include "tq2e.cin"
// #include "tq2L.cin"
#include "sqrt2f21L.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
 int M=211,M1=M+1;
 int N=201,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("sqrt2f21lma.eps","w");  ado(o,0,0,214,212);
 fprintf(o,"112 110 translate\n 10 10 scale\n");
// DB sy=10.1/sinh(N/2./100.);
 DO(m,M1) X[m]=-11+.1*(m-.5);
 DO(n,N1) Y[n]=-10+.1*(n-.5);
// DO(n,N1) Y[n]=sy*sinh((n-N/2.+.5)/100.);
 for(m=-10;m<11;m++) {M(m,-10)L(m,10)}
 for(n=-10;n<11;n++) {M(  -10,n)L(10,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
/*
 fprintf(o,"/adobe-Roman findfont 1 scalefont setfont\n");
 for(m=-8;m<0;m+=2) {M(-11.2,m-.3) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<9;m+=2) {M(-10.7,m-.3) fprintf(o,"(%1d)s\n",m);}
 for(m=-8;m<0;m+=4) {M(m-.6,-10.8) fprintf(o,"(%1d)s\n",m);}
 for(m= 0;m<9;m+=4) {M(m-.3,-10.8) fprintf(o,"(%1d)s\n",m);}
 fprintf(o,"/Times-Italic findfont 1 scalefont setfont\n");
 //fprintf(o,"/adobe-italic findfont 1 scalefont setfont\n");
 M(  9.6,-10.8) fprintf(o,"(y)s\n");
 M(-10.7,  9.5) fprintf(o,"(x)s\n");
// M(-11,0)L(10.1,0) M(0,-11)L(0,10.1) fprintf(o,".01 W 1 0 1 RGB S\n");
// z_type tm,tp,F[M1*N1];; 
*/
 DO(m,M1)DO(n,N1){      g[m*N1+n]=9999;
                        f[m*N1+n]=9999;}
 DO(m,M1){x=X[m];
 DO(n,N1){y=Y[n]; z=z_type(x,y);        
        if( abs(z-2.)>.2 || abs(z-4.)>.2)
        {       c=F21L(z);
                p=Re(c);  q=Im(c);
                if(p>-99 && p<99) g[m*N1+n]=p;
                if(q>-99 && q<99 && fabs(q)>1.e-18) f[m*N1+n]=q;
        }}}
//p=2; q=1.1;
 #include "plofu.cin"

// M(-13.2,0)L(2,0) fprintf(o,".05 W 0 .8 0 RGB S\n");

M(2,0)L(10.1,0)fprintf(o,".05 W 1 1 1 RGB S\n");
DO(n,27){M(2+.3*n,0)L(2+.3*(n+.5) ,0)} fprintf(o,".1 W 0 0 0 RGB S\n");

//M(2,0)L(10.1,0)fprintf(o,".1 W 0 0 0 RGB [.19 .19] 0 setdash S\n");
//M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n"); // fails at some printers
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf sqrt2f21lma.eps"); 
        system(    "open sqrt2f21lma.pdf"); //for linux
//      getchar(); system("killall Preview"); // For macintosh
}

Latex generator of the labels

\documentclass[12pt]{article}
\paperwidth 422px
\paperheight 418px
\textwidth 1394px
\textheight 1300px
\topmargin -94px
\oddsidemargin -76px
\usepackage{graphics}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \rmi {\mathrm{i}}
\parindent 0pt
\pagestyle{empty}
\begin{document}\parindent 0pt

\sx{2}{\begin{picture}(204,204) \put(0,0){\ing{sqrt2f21lma}}
\put(6,206){\sx{.8}{$y$}}
\put(6,188){\sx{.8}{$8$}}
\put(6,168){\sx{.8}{$6$}}
\put(6,148){\sx{.8}{$4$}}
\put(6,128){\sx{.8}{$2$}}
\put(6,108){\sx{.8}{$0$}}
\put(-1, 88){\sx{.8}{$-2$}}
\put(-1, 68){\sx{.8}{$-4$}}
\put(-1, 48){\sx{.8}{$-6$}}
\put(-1, 28){\sx{.8}{$-8$}}
\put(24,2){\sx{.8}{$-8$}}
\put(44,2){\sx{.8}{$-6$}}
\put(64,2){\sx{.8}{$-4$}}
\put(84,2){\sx{.8}{$-2$}}
\put(110.5,2){\sx{.8}{$0$}}
\put(130.5,2){\sx{.8}{$2$}}
\put(150.5,2){\sx{.8}{$4$}}
\put(170.5,2){\sx{.8}{$6$}}
\put(190.5,2){\sx{.8}{$8$}}
\put(208.6,2){\sx{.8}{$x$}}

\put(26,198){$v\!=\!0$}
\put(18,152){\rot{-2}$u\!=\!-2$\ero}
\put(26,107.4){$v\!=\!0$}  \put(176,108){\bf cut}
\put(18,62){\rot{1}$u\!=\!-2$\ero}
\put(26,17){$v\!=\!0$}

\put(96,172){\rot{20}$v\!=\!0.2$\ero}
\put(106,158){\rot{20}$v\!=\!0.4$\ero}
\put(112,148){\rot{20}$v\!=\!0.6$\ero}
\put(121,139){\rot{20}$v\!=\!1$\ero}

\put(156,167){\rot{37}$u\!=\!-3$\ero}
\put(178,164){\rot{50}$v\!=\!0$\ero}

\put(74,95){\rot{-23}$u\!=\!-1.8$\ero}
\put(87,28){\rot{54}$u\!=\!-2.2$\ero}
\put(113,21){\rot{76}$u\!=\!-2.4$\ero}
\put(132,21){\rot{87}$u\!=\!-2.6$\ero}
\put(137,64){\rot{-79}$u\!=\!-2.8$\ero}
\put(151,51){\rot{-43}$u\!=\!-3$\ero}

\put(175,51){\rot{-48}$v\!=\!0$\ero}
\end{picture}}
\end{document}

References

Keywords

«[[]]», «Abel function», «Arctetration», «Base sqrt2», «Book», «BookMap», «C++», «Complex map», «Exp», «Exponential», «Latex», «Lambert Academic Publishing», «Superfunction», «Superfunctions», «Tetration», «ado.cin», «conto.cin», «sqrt2f21e.cin»,

«Суперфункции», «Тетрация»,