File:Vladi11.jpg

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Fig.15.3 at page 208 of book «Superfunctions» ^[1], 2020.

It appears also as Рис.15.3 at page 209 of the Russian version «Суперфункции» ^[2], 2020.

The pictures show the agreements at the substitution of Agreements at the substitution of values by numerical implementations FSE of natural tetration tet and the implementation FSL of the natural arctetration ate into equations

\(\mathrm{ate}(\mathrm{tet}(z))=z\)

\(\mathrm{tet}(\mathrm{ate}(z))=z\)

Tetration is implemented as FSE and FSL

Arctetration is implemented as FSL.

The agreements are \[ \displaystyle D_{\rm at}(z)= -\ln\left( \frac {|\mathrm{FSL}(\mathrm{FSE}(z)) - z|} {|\mathrm{FSL}(\mathrm{FSE}(z))| + |z|} \right) \]

\[ \displaystyle D_{\rm ta}(z)= -\ln\left( \frac {|\mathrm{FSE}(\mathrm{FSL}(z)) - z|} {|\mathrm{FSE}(\mathrm{FSL}(z))| + |z|} \right) \]

Map \(D=D_{\rm at}(x\!+\!\mathrm i y)\) is shown at left, and map of \(D=D_{\rm at}(x\!+\!\mathrm i y)\) is shown at right.

Similar maps are published in the Vladikavkaz Matehmatical Journal ^[3].

Warning: In the code below, function FSE is demoted with identifier "FSEXP" and function FSL is denoted with identifier "FSLOG".

C++ generator of the left hand side map

/* ado.cin, conto.cin, fslog.cin fsexp.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;
//#include <complex.h>
//#define z_type complex<double>
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)

//#include"superex.cin"
//#include"superlo.cin"
#include"fsexp.cin"
#include"fslog.cin"
#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
z_type Zo=z_type(.31813150520476413, 1.3372357014306895);
z_type Zc=z_type(.31813150520476413,-1.3372357014306895);

int M=200,M1=M+1;
int N=300,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("figslosexp.eps","w");ado(o,0,0,404,604);
FILE *o;o=fopen("vladi11a.eps","w");ado(o,404,604);
fprintf(o,"202 302 translate\n 50 50 scale\n");

DO(m,M1) X[m]=-4+.04*m;
DO(n,N1) Y[n]=-6+.04*(n+.5);

for(m=-4;m<5;m++) {     if(m==0){M(m,-6.1)L(m,6.1)}
                        else    {M(m,-6)L(m,6)}                 }
for(n=-6;n<7;n++) {M(  -4,n)L(4,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]; printf("50 run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
        c=FSEXP(z);
        if(abs(c)<1.e128){      c=FSLOG(c);
                                d=z;
                                p=abs(c-d);
                                p=-log(p)/log(10.);
                        if(p>-99 && p<99) g[m*N1+n]=p;
                        }
        }}

p=40;q=8;
#include "plodi.cin"
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi11a.eps");
        system(    "open vladi11a.pdf");// linux
//getchar(); system("killall Preview");//macintosh
}

C++ generator of the right hand side map

/* ado.cin, conto.cin, fslog.cin, fsexp.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"superex.cin"
//#include"superlo.cin"
#include"fsexp.cin"
#include"fslog.cin"
#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
z_type Zo=z_type(.31813150520476413, 1.3372357014306895);
z_type Zc=z_type(.31813150520476413,-1.3372357014306895);

int M=200,M1=M+1;
int N=300,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("figslosexp.eps","w");ado(o,0,0,404,604);
FILE *o;o=fopen("vladi11b.eps","w");ado(o,404,604);
fprintf(o,"202 302 translate\n 50 50 scale\n");

DO(m,M1) X[m]=-4+.04*m;
DO(n,N1) Y[n]=-6+.04*(n+.5);

for(m=-4;m<5;m++) {     if(m==0){M(m,-6.1)L(m,6.1)}
                        else    {M(m,-6)L(m,6)}                 }
for(n=-6;n<7;n++) {M(  -4,n)L(4,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]; printf("50 run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
        c=FSLOG(z);
        if(abs(c)<1.e128){      c=FSEXP(c);
                                d=z;
                                p=abs(c-d);
                                p=-log(p)/log(10.);
                        if(p>-99 && p<99) g[m*N1+n]=p;
                        }
        }}

p=40;q=8;
#include "plodi.cin"

//M(Re(Zo),Im(Zo)) L(-2.02,Im(Zo)) M(Re(Zo),-Im(Zo)) L(-2.02,-Im(Zo)) 
//fprintf(o,".08 W 1 1 1 RGB S\n");
DO(m,14){ M(Re(Zo)-.31*(m  ), Im(Zo)) L(Re(Zo)-.31*(m+.5), Im(Zo)) }
DO(m,14){ M(Re(Zo)-.31*(m  ),-Im(Zo)) L(Re(Zo)-.31*(m+.5),-Im(Zo)) }
fprintf(o,"0 setlinecap .08 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi11b.eps");
        system(    "open vladi11b.pdf");// linux
//getchar(); system("killall Preview");//macintosh
}

Latex combiner

\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{geometry}
\paperwidth 418px
%\paperheight 134px 
\paperheight 296px 
\topmargin -96pt
\oddsidemargin -94pt
\pagestyle{empty}
\begin{document}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}

\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}


\newcommand \vladiEL {
\put(-18,590){\sx{2.3}{$y$}}
\put(-18,544){\sx{2.3}{4}}
\put(-18,494){\sx{2.3}{4}}
\put(-18,444){\sx{2.3}{3}}
\put(-18,394){\sx{2.3}{2}}
\put(-18,344){\sx{2.3}{1}}
\put(-18,294){\sx{2.3}{0}}
\put(-36,244){\sx{2.3}{$-1$}}
\put(-36,194){\sx{2.3}{$-2$}}
\put(-36,144){\sx{2.3}{$-3$}}
\put(-36, 94){\sx{2.3}{$-4$}}
\put(-36, 44){\sx{2.3}{$-5$}}
%\put(-20, -20){\sx{2.3}{$-4$}}
\put( 30, -20){\sx{2.3}{$-3$}}
\put( 80, -20){\sx{2.3}{$-2$}}
\put(130, -20){\sx{2.3}{$-1$}}
\put(196, -20){\sx{2.3}{$0$}}
\put(246, -20){\sx{2.3}{$1$}}
\put(296, -20){\sx{2.3}{$2$}}
\put(346, -20){\sx{2.3}{$3$}}
\put(391, -20){\sx{2.3}{$x$}}
}

%\begin{figure}[t]
%\begin{center}
\vskip 16pt
\hskip 16pt
%\sx{.54}{\begin{picture}(472,580)
\sx{.47}{\begin{picture}(456,580)
%\put(0,0){\includegraphics{figslosexp}}
\put(0,0){\includegraphics{vladi11a}}
\vladiEL
\put(110,520){\sx{2.8}{$10\!<\!D\!<\!12$}}
\put(154,420){\sx{2.4}{$12\!<D\!<\!14$}}
\put(160,310){\sx{2.8}{$D\!>\!14$}}
\put(154,168){\sx{2.4}{$12\!<D\!<\!14$}}
\put(110,64){\sx{2.8}{$10\!<\!D\!<\!12$}}
\put(20, 320){\sx{2.8}{$D\!<\!1$}}
\put(310,560){\sx{2.8}{$D\!<\!1$}}
\put(310,20){\sx{2.8}{$D\!<\!1$}}
\end{picture}}
\sx{.47}{\begin{picture}(388,580)
%\put(0,0){\includegraphics{figsexpslo}}
\put(0,0){\includegraphics{vladi11b}}
\vladiEL
\put( 20,379){\sx{2.5}{$12\!<\!D\!<\!14$}}
\put( 20,210){\sx{2.5}{$12\!<\!D\!<\!14$}}
\put(200,310){\sx{3}{$D\!>\!14$}}
%\put(204,373){\sx{1}{11}}
%\put(204,226){\sx{1}{11}}
\end{picture}}
\end{document}

References

↑ https://mizugadro.mydns.jp/BOOK/408.pdf Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.
↑ https://mizugadro.mydns.jp/BOOK/202.pdf Дмитрий Кузнецов. Суперфункции. Lambert Academic Publishing, 2020.
↑ http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45. Figure 11.

Keywords

«Abel function», «Abelfunction», «Agreement», «Arctetration», «Book», «BookMap», «C++», «Complex map», «Exp», «Exponential», «Latex», «Natural tetration», «Superfunctions», «Test», «Tetration»,

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[1] ttps://mizugadro.mydns.jp/BOOK/408.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://mizugadro.mydns.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45. Figure 11.

[1]

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