Difference between revisions of "File:Sqrt2q2map600.jpg"

Fig.16.7 from page 227 of book «Superfunctions»^[1], 2020.

This picture is also used as Рис. 16.8 at page 233 of the Russian version «Суперфункции» ^[2], 2014.

The same map appears also at the top left picture of figure 6 of article ^[3] at Mathematics of Computation, 2010.

The figure shows the Complex map of the half iterate of exponent to base \(\sqrt{2}\) regular at its lowest fixed point: \[ u\!+\!\mathrm i v= \exp_{\sqrt{2},\mathrm d}^{~ 1/2}(x\!+\!\mathrm i y) \]

This function is expressed through tetration and arctetration to base \(\sqrt{2}\): \[ \varphi(z)=\exp_{\sqrt{2},\mathrm d}^{~ 1/2}(z)= \mathrm{tet}_{\sqrt{2}}\left( \frac{1}{2}+\mathrm{ate}_{\sqrt{2}}(z) \right) \]

Function \(\varphi\) is solution of equation \[ \varphi(\varphi(z))=2^z \]

For natural exponential, id est, for base \(b\!=\!\mathrm e\), such a function \(\varphi\) has been announced by Hellmuth Kneser ^[4] in 1950 and reported ^[5][6] in 2009, 2010; it satisfies equation \(\varphi(\varphi(z))=\mathrm e^z\).

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++)
 #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 "uq2e.cin"
// #include "uq2L.cin"
// #include "f43E.cin"
// #include "f43L.cin"
//#include "sqrt2f23e.cin"
//#include "sqrt2f23l.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=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("sqrt26a.eps","w");  ado(o,214,212);
 fprintf(o,"112 110 translate\n 10 10 scale\n");
 DO(m,M1) X[m]=-11+.1*(m-.5);
 DO(n,N1) Y[n]=-10+.1*(n-.5);
 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(-10.7,  9.5) fprintf(o,"(y)s\n");
// M(  9.6,-10.8) 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-10.)>2.9)    
        {
 //     c=UQ2E(z) ; d=uq2LA(c);
 //     c=UQ2L(z); c=UQ2E(c+.5);
 //     c=F43L(z); c=F43E(c+.5);
        c=F23L(z); c=F23E(c+.5);
  //    p=abs((z-d)/(z+d)); p=-log(p)/log(10.); 
        p=Re(c);  q=Im(c);
        if(p>-99 && p<99 && fabs(p)>1.e-12) g[m*N1+n]=p;
        if(q>-99 && q<99 && fabs(q)>1.e-12) f[m*N1+n]=q;
        }}}
p=2; q=1;
#include "plofu.cin"

// p=1.e-15;
//for(n=-10;n<11;n++) {q=p*n; z=z_type(q,0.); printf("%19.15f %19.16f\n",q,  Re(uq2e(z))  ); }
//M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n");

/*
y=9.064720284;
M(.89, y)L(10.1, y)
M(.89,-y)L(10.1,-y)
M(4,0)L( 10,0)fprintf(o,".15 W 0 0 0 RGB [.2 .2] 0 setdash S\n");
*/
//The cuts live in a separate file, but one will be more stressed:
M(4,0)L( 10.1,0)
fprintf(o,".1  W 1 1 1 RGB S\n");
DO(n,28){M(4+.3*n,0)L(4+.3*(n+.5),0)} fprintf(o,".13 W 0 0 0 RGB S\n");

 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf sqrt26a.eps"); 
        system(    "open sqrt26a.pdf"); // for LINUX 
//      getchar(); system("killall Preview"); // For macintosh
}

C++ generator of the cut lines

/* 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++)
 #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 "uq2e.cin"
// #include "uq2L.cin"
// #include "f43E.cin"
// #include "f43L.cin"

// #include "f23E.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= 60,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("sqrt26cuts.eps","w");  ado(o,214,212);
 fprintf(o,"112 110 translate\n 10 10 scale\n");
 DO(m,M1) X[m]= 4.2  +.1*(m-.5);
 DO(n,N1) Y[n]=-10+.1*(n-.5);
 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(-10.7,  9.5) fprintf(o,"(y)s\n");
// M(  9.6,-10.8) 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];
        if(fabs(fabs(y)-9.)>.3)
        {        z=z_type(x,y); 
        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-12) f[m*N1+n]=q;
        }}}
//#include "plofu.cin"

conto(o,f,w,v,X,Y,M,N, (0.      ), -2,2); // fprintf(o,".05 W 0 .8 0 RGB S\n");

/*
y=9.064720284;
M(.89, y)L(10.1, y)
M(.89,-y)L(10.1,-y)
M(4,0)L( 10,0)
*/

// fprintf(o,".1 W 0 0 0 RGB [.2 .2] 0 setdash S\n");

y=9.064720284;
M(.89, y)L(10.1, y)
M(.89,-y)L(10.1,-y)
M(4,0)L( 10,0)fprintf(o,".07 W 0 0 0 RGB [.2 .2] 0 setdash S\n");

fprintf(o,"[99]0 setdash\n");

/*
M(2,0)L(-11.2,0)fprintf(o,".1  W 1 1 1 RGB S\n");
DO(n,36){M(2-.3*n,0)L(2-.3*(n+.5),0)} fprintf(o,".15 W 0 0 0 RGB S\n");
//M(2,0)L(-11,0)fprintf(o,".15 W 0 0 0 RGB [.2 .2] 0 setdash S\n"); //may cause problems
*/

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf sqrt26cuts.eps"); 
        system(    "open sqrt26cuts.pdf"); 
//      getchar(); system("killall Preview"); // For macintosh
}

Latex combiner

\documentclass[12pt]{article}
\paperwidth 212px
\paperheight 210px
\textwidth 1394px
\textheight 1300px
\topmargin -100px
\oddsidemargin -74px
\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{1.}{\begin{picture}(280,203)
%\put(0,0){\ing{exc2cuts}}
%\put(0,0){\ing{figexc2}}
\put(0,0){\ing{sqrt26cuts}}
\put(0,0){\ing{sqrt26a}}
\put(6,207.7){\sx{.7}{$y$}}
\put(6,187.8){\sx{.7}{$8$}}
\put(6,167.8){\sx{.7}{$6$}}
\put(6,147.7){\sx{.7}{$4$}}
\put(6,127.7){\sx{.7}{$2$}}
\put(6,107.7){\sx{.7}{$0$}}
\put(0, 87.7){\sx{.7}{$-2$}}
\put(0, 67.6){\sx{.7}{$-4$}}
\put(0, 47.6){\sx{.7}{$-6$}}
\put(0, 27.5){\sx{.7}{$-8$}}
\put(3, 2){\sx{.7}{$-10$}}
\put( 27,2){\sx{.7}{$-8$}}
\put( 47,2){\sx{.7}{$-6$}}
\put( 67,2){\sx{.7}{$-4$}}
\put( 87,2){\sx{.7}{$-2$}}
\put(111,2){\sx{.7}{$0$}}
\put(131,2){\sx{.7}{$2$}}
\put(151,2){\sx{.7}{$4$}}
\put(171,2){\sx{.7}{$6$}}
\put(191,2){\sx{.7}{$8$}}
\put(210,2){\sx{.7}{$x$}}
%\put( 27,127){\sx{3}{$\exp_{b,2}^{[1/2]}$}}
\put(23,198.6){\sx{.9}{$v\!=\!0$}}
\put(33.7,92){\rot{90}\sx{.9}{$u\!=\!-1.2$}\ero}
\put(59.6,96){\rot{90}\sx{.9}{$u\!=\!-1$}\ero}
\put(102,98){\rot{90}\sx{.9}{$u\!=\!0$}\ero}
\put(122,99){\rot{90}\sx{.9}{$u\!=\!1$}\ero}
\put(136,99){\rot{90}\sx{.9}{$u\!=\!2$}\ero}
%\put(215,207){\sx{.8}{\bf cut}}
\put(195,199){\sx{.8}{\bf cut}}
\put(197,184){\rot{24}{\sx{.78}{\bf cut}}\ero}
\put(204.4,174.4){\rot{40}{\sx{.78}{\bf cut}}\ero}
%\put(215,184){\sx{.8}{\bf cut}}
%\put(215,175){\sx{.8}{\bf cut}}
\put(195,108){\sx{.78}{\bf cut}}
%
\put(87,145){\rot{-44}\sx{.9}{$v\!=\!1$}\ero}
\put(61,122){\rot{-15}\sx{.8}{$v\!=\!0.2$}\ero}
\put(63,108){\sx{.9}{$v\!=\!0$}}
\put(62, 94){\rot{13}\sx{.8}{$v\!=\!-0.2$}\ero}
%\put(215, 42){\sx{.8}{\bf cut}}
%\put(215, 33){\sx{.8}{\bf cut}}
%\put(215, 25){\sx{.8}{\bf cut}}
\put(196,32){\rot{-22}{\sx{.78}{\bf cut}}\ero}
\put(195, 17){\sx{.78}{\bf cut}}
%\put(-23, 17){\sx{1.}{$q\!=0$}}
%\put(215, 10){\sx{.8}{\bf cut}}
\put(23,17){\sx{.9}{$v\!=\!0$}}
\end{picture}}

\end{document}

References

Keywords

«Base sqrt2», «Complex map», «Exponential», «Superfunction», «Superfunctions», «Tetration»,

«Суперфункции»,