File:Sqrt2diimap80.jpg

Fig.16.10 at page 234 of book «Superfunctions»^[1]

The same image appears also as Рис.16.11 at page 240 of the Russian version «Суперфункции»^[2], 2014.

The picture shows the complex map of iterate number i of exponential to base \(\sqrt{2}\) constructed at its lower ("down") fixed point 2: \[ u\!+\!\mathrm i v= \exp_{\sqrt{2},\mathrm d}(x\!+\!\mathrm i y) \]

The algorithm of the evaluation is also described in «Mathematics of computation»^[3], 2010.

C++ generator of the map

/* Files ado.cin, conto.cin, sqrt2f21e.cin sqrt2f21l.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++)
// using namespace std;
 #include <complex>
 typedef std::complex<double> z_type;
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #include "conto.cin"

#include "sqrt2f45e.cin"
#include "sqrt2f45l.cin"
#include "sqrt2f21e.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=801,M1=M+1;
 int N=405,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("04.eps","w");  ado(o,202,202);
 FILE *o;o=fopen("sqrt2diima.eps","w");  ado(o,202,202);
 fprintf(o,"101 101 translate\n 10 10 scale\n");
 DO(m,M1) X[m]=-10.+.025*(m-.5);
 //DO(n,N1) Y[n]=-10.+.04*(n-.5);
 // DO(n,200) Y[n]=sinh(3.*(n-200.5)/200.);
//  DO(n,200) Y[n]=-10.+.05*(n-.5);
//         Y[200]=-.0001;
//         Y[201]= .0001;
for(n=0;n<N1;n++) Y[n]=.25*sinh(4.33*(n-202.5)/200.);
// for(n=202;n<N1;n++) Y[n]=-10.+.05*(n-2);
 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,"1 setlinejoin 2 setlinecap\n");
 fprintf(o," .006 W 0 0 0 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); 
                c=F21L(z) + z_type(0.,1.); c=F21E(c);
                p=Re(c); 
                q=Im(c);
                if(p>-201. && p<201. && q>-201. && q<201.
               && fabs(p)>1.e-14
               && fabs(q)>1.e-14
                ) { g[m*N1+n]=p; f[m*N1+n]=q;}
        }}
 p=2; 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,".014 W 0 .7 0 RGB S\n");
 for(m=0;m<11;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);                  fprintf(o,".014 W 1 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,".014 W 0 0 1 RGB S\n");
 for(m= 1;m<40;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".03 W .8 0 0 RGB S\n");
 for(m= 1;m<40;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".03 W 0 0 .8 RGB S\n");
                conto(o,f,w,v,X,Y,M,N, (0.  ),-2*p,2*p); fprintf(o,".03 W .5 0 .5 RGB S\n");
 for(m=-40;m<41;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n");

 // #include "plofu.cin"
  M(10,0)L(4,0)fprintf(o,"0 setlinecap .036 W 1 1 1 RGB S\n");
 for(n=0;n<17;n++){ M(4+.5*(n+.2),0) L(4+.5*(n+.4),0) } fprintf(o,".06 W 1 .5 0 RGB S\n");
 for(n=0;n<17;n++){ M(4+.5*(n+.7),0) L(4+.5*(n+.9),0) } fprintf(o,".06 W 0 .5 1 RGB S\n");
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf sqrt2diima.eps"); 
        system(    "open sqrt2diima.pdf"); //for macintosh
 }

Latex generator of labels

%Files generated with codes above %should be loaded in order to compile the code below.

\documentclass[12pt]{article}
\paperwidth 2072px
\paperheight 2076px
\textwidth 2394px
\textheight 2300px
\topmargin -97px
\oddsidemargin -78px
\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{10}{\begin{picture}(206,206)
%\put(6,5){\ing{Esqrt2ite13Map}}
%\put(6,5){\ing{04}}
\put(6,5){\ing{sqrt2diima}}
\put(2,203.4){\sx{.7}{$y$}}
\put(2,184){\sx{.6}{$8$}}
\put(2,164){\sx{.6}{$6$}}
\put(2,144){\sx{.6}{$4$}}
\put(2,124){\sx{.6}{$2$}}
\put(2,104){\sx{.6}{$0$}}
\put(-2.2,84){\sx{.6}{$-2$}}
\put(-2.2,64){\sx{.6}{$-4$}}
\put(-2.2,44){\sx{.6}{$-6$}}
\put(-2.2,24){\sx{.6}{$-8$}}
\put(-2,-1){\sx{.7}{$-\!10$}}
\put( 22,-1){\sx{.7}{$-8$}}
\put( 42,-1){\sx{.7}{$-6$}}
\put( 62,-1){\sx{.7}{$-4$}}
\put( 82,-1){\sx{.7}{$-2$}}
\put(106,-1){\sx{.7}{$0$}}
\put(126,-1){\sx{.7}{$2$}}
\put(146,-1){\sx{.7}{$4$}}
\put(166,-1){\sx{.7}{$6$}}
\put(186,-1){\sx{.7}{$8$}}
\put(204,-1){\sx{.7}{$x$}}
\put(174,103.5){\sx{.99}{\bf cut}}
\put(118,172.3){\sx{.99}{\rot{-16}$v\!=\!3$\ero}}
%\put(146,135){\sx{.99}{\rot{-29}$v\!=\!2$\ero}}
%\put(148,117){\sx{.99}{\rot{-33}$v\!=\!1$\ero}}
\put(125,139){\sx{.99}{\rot{-11}$v\!=\!2$\ero}}
\put(126,118){\sx{.99}{\rot{-3}$v\!=\!1$\ero}}
\put(121,103){\sx{.99}{\rot{6}$v\!=\!0$\ero}}
\put(119,94){\sx{.99}{\rot{3}$v\!=\!-1$\ero}}
%
\put(27,140){\sx{.99}{\rot{-30}$u\!=\!1$\ero}}
\put(91,91){\sx{.99}{\rot{7}$u\!=\!0$\ero}}
%
\put(117,180){\sx{.99}{\rot{74}$u\!=\!2$\ero}}
\put(155,168){\sx{.99}{\rot{58}$u\!=\!3$\ero}}
\put(155,120){\sx{.99}{\rot{55}$u\!=\!4$\ero}}
\end{picture}}
\end{document}

Refereces

Keywords

«Abelfunction», «Arctetration», «Base sqrt2», «Complex map», «Iterate», «Inverse function», «Lambert Academic Publishing», «Superfunction», «Superfunctions», «Tetration»,

«Суперфункции»,