File:Sqrt2tetatemap.jpg

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Fig.16.4 from page 224 of book «Superfunctions» ^[1], 2020.

This image appears also as Рис.16.4 at page 229 of the Russian version «Суперфункции» ^[2], 2020.

This image is used also in figure 2 of article ^[3] at Mathematics of Computation, 2010 (top right map)

The image shows the range of validity of relation

\(\mathrm{tet}_{\sqrt{2}}(\mathrm{ate}_{\sqrt{2}}(z))=z\)

for the tetration and arctetration to base \(\sqrt{2}\) in the plane \(x=\Re(z)\), \(y=\Im(z)\) is shaded with lines

\(u=\Re \Big( \mathrm{tet}_{\sqrt{2}}(\mathrm{ate}_{\sqrt{2}}(z)) \Big) = \mathrm{const}\)

and lines

\(v=\Im \Big( \mathrm{tet}_{\sqrt{2}}(\mathrm{ate}_{\sqrt{2}}(z)) \Big) = \mathrm{const}\)

These lines provide the shading of the central part of the picture.

In addition, lines

\(\Im\Big( \exp_{\sqrt{2}}^n (x\!+\!\mathrm i y)\Big)=\pm \frac{|P|}{2}=\pm \frac{2\, \pi}{\ln(2)}\)

are drawn for \(n=0,1,2,3,4\) as bounds if the shaded range. Here \(P=4 \pi \mathrm i / \ln(2)\) is period of Tetration to base \(\sqrt{2}\).

C++ generator of the shading

/* 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 "tq2e.cin"
// #include "tq2L.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=401,M1=M+1;
 int N=501,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("sqrt2tetatema.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]=-10+.05*(m-.5);
 DO(n,N1) Y[n]=-10+.04*(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");
 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);
                c=F21E(c);
                if(abs(c-z)<.1) {
                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;
                }
        }}}
fprintf(o,"1 setlinejoin 2 setlinecap\n");
//p=2; q=1.1;
p=.1;q=.1;
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,".01 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,".01 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,".01 W 0 0 .9 RGB S\n");

for(m=1;m<11;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".06 W .9 0 0 RGB S\n");
for(m=1;m<11;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".06 W 0 0 .9 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p);fprintf(o,".06 W .6 0 .6 RGB S\n");
for(m=-10;m<11;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".06 W 0 0 0 RGB S\n");


//#include "plofu.cin"

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

fprintf(o,"0 setlinejoin 0 setlinecap\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 sqrt2tetatema.eps"); 
        system(    "open sqrt2tetatema.pdf"); //for linux
//      getchar(); system("killall Preview"); // For macintosh
}

C++ generator boundaries of the shading

/* 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 "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=201,M1=M+1;
 int N=401,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("sqrt2tetatemb.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]=0+.05*(m-.5);
 DO(n,N1) Y[n]=-10+.05*(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");

 DO(m,M1)DO(n,N1){      g[m*N1+n]=9999;
                        f[m*N1+n]=9999;}

M(-10, 2./log(2.)*M_PI) L( 10, 2./log(2.)*M_PI)
M(-10,-2./log(2.)*M_PI) L( 10,-2./log(2.)*M_PI)

p=96.;
 DO(m,M1){x=X[m];
 DO(n,N1){y=Y[n]; z=z_type(x,y);        
                c=exp(.5*log(2.)*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;
                  conto(o,f,w,v,X,Y,M,N, (2./log(2.)*M_PI  ),-p,p);fprintf(o,".06 W .6 0 .6 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (-2./log(2.)*M_PI  ),-p,p);fprintf(o,".06 W .6 0 .6 RGB S\n");

 DO(m,M1){x=X[m];
 DO(n,N1){y=Y[n]; z=z_type(x,y);        
                c=exp(.5*log(2.)*z);
                c=exp(.5*log(2.)*c);
                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=96.;
                  conto(o,f,w,v,X,Y,M,N, (2./log(2.)*M_PI  ),-p,p);fprintf(o,".06 W .6 0 0 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (-2./log(2.)*M_PI  ),-p,p);fprintf(o,".06 W .6 0 0 RGB S\n");

 DO(m,M1){x=X[m];
 DO(n,N1){y=Y[n]; z=z_type(x,y);        
                c=exp(.5*log(2.)*z);
                c=exp(.5*log(2.)*c);
                c=exp(.5*log(2.)*c);
                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=96.;
                  conto(o,f,w,v,X,Y,M,N, (2./log(2.)*M_PI  ),-p,p);fprintf(o,".06 W 0 0 .6 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (-2./log(2.)*M_PI  ),-p,p);fprintf(o,".06 W 0 0 .6 RGB S\n");

 DO(m,M1){x=X[m];
 DO(n,N1){y=Y[n]; z=z_type(x,y);        
                c=exp(.5*log(2.)*z);
                c=exp(.5*log(2.)*c);
                c=exp(.5*log(2.)*c);
                c=exp(.5*log(2.)*c);
                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=96.;
                  conto(o,f,w,v,X,Y,M,N, (2./log(2.)*M_PI  ),-p,p);fprintf(o,".06 W .0 .5 0 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (-2./log(2.)*M_PI  ),-p,p);fprintf(o,".06 W 0 .5 0 RGB S\n");

 DO(m,M1){x=X[m];
 DO(n,N1){y=Y[n]; z=z_type(x,y);        
                c=exp(.5*log(2.)*z);
                c=exp(.5*log(2.)*c);
                c=exp(.5*log(2.)*c);
                c=exp(.5*log(2.)*c);
                c=exp(.5*log(2.)*c);
                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=96.;
                  conto(o,f,w,v,X,Y,M,N, (2./log(2.)*M_PI  ),-p,p);fprintf(o,".06 W .0 0 0 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (-2./log(2.)*M_PI  ),-p,p);fprintf(o,".06 W 0 0 0 RGB S\n");

 DO(m,M1){x=X[m];
 DO(n,N1){y=Y[n]; z=z_type(x,y);        
                c=exp(.5*log(2.)*z);
                c=exp(.5*log(2.)*c);
                c=exp(.5*log(2.)*c);
                c=exp(.5*log(2.)*c);
                c=exp(.5*log(2.)*c);
                c=exp(.5*log(2.)*c);
                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=96.;
                  conto(o,f,w,v,X,Y,M,N, (2./log(2.)*M_PI  ),-p,p);fprintf(o,".06 W .0 0 0 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (-2./log(2.)*M_PI  ),-p,p);fprintf(o,".06 W 0 0 0 RGB S\n");

/*
p=4;q=1;
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,".01 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,".01 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,".01 W 0 0 .9 RGB S\n");

for(m=1;m<11;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".06 W .9 0 0 RGB S\n");
for(m=1;m<11;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".06 W 0 0 .9 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p);fprintf(o,".06 W .6 0 .6 RGB S\n");
for(m=-10;m<11;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".06 W 0 0 0 RGB S\n");
*/

// #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");

 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf sqrt2tetatemb.eps"); 
        system(    "open sqrt2tetatemb.pdf"); //for linux
//      getchar(); system("killall Preview"); // For macintosh
}

</nowiki></nomathjax></poem>

==[[Latex]] combiner==
Files generated with codes above
should be loaded in order to compile the code below.

<poem><nomathjax><nowiki>
\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{sqrt2tetatema}}
\put(0,0){\ing{sqrt2tetatemb}}
\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(26,107.4){$v\!=\!0$} 
%\put(176,108){\bf cut}
\put(115.4,57.4){\rot{90}$u\!=\!0$\ero} 
\end{picture}}
\end{document}

Refereces

↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.
↑ https://mizugadro.mydns.jp/BOOK/202.pdf Дмитрий Кузнецов. Суперфункции. Lambert Academic Publishing, 2020.
↑ http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html
http://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf offprint D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.

Keywords

«[[]]», «[[]]», «[[]]»,

«Agreement», «Arctetration», «Base sqrt2», «Book», «BookMap», «C++», «Complex map», «Identity function», «Inverse function», «Latex», «Mathematics of Computation», «Superfunction», «Superfunctions», «Tetration»,

«Тетрация»,

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	Date/Time	Thumbnail	Dimensions	User	Comment
current	06:14, 1 December 2018		1,758 × 1,741 (1,008 KB)	Maintenance script (talk \| contribs)	Importing image file

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Horizontal resolution	300 dpi
Vertical resolution	300 dpi

[1] ttps://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.

[2] ttps://mizugadro.mydns.jp/BOOK/202.pdf Дмитрий Кузнецов. Суперфункции. Lambert Academic Publishing, 2020.

[3] ttp://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html
http://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf offprint D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.

[1]

[2]

[3]