File:Vladi06.jpg

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Original file ‎(1,741 × 791 pixels, file size: 418 KB, MIME type: image/jpeg)

Complex map of the approximation "tai" of natural tetration, left, and the map of the corresponding agreement function $D$.

Left:

$u\!+\!\mathrm i v = \mathrm{tai}(x+\mathrm i y)$

Right:

$\displaystyle D_6=D_6(z)=-\lg\left( \frac

$ levels $D=1,2,4,6,8,10,12,14 ~ ~ $ are drawn. Level $D=1$ is drawn with thick line. Usage: this is figure 14.8 of the book Суперфункции (2014, In Russian) [1]
the English version is in preparation in 2015.
First time published in the Vladikavkaz Matehmatical Journal[2].

C++ generator of the first picture

fsexp.cin, ado.cin, conto.cin should be loaded in order to compile the code below


#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()
#define I z_type(0.,1.)

#include "fsexp.cin"
//#include "superex.cin"
//#include "superlo.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=201,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("figtai3.eps","w");ado(o,0,0,82,82);
FILE *o;o=fopen("vladi06a.eps","w");ado(o,82,82);
fprintf(o,"41 11 translate\n 10 10 scale\n");

DO(m,M1) X[m]=-4.+.04*(m-.5);
//DO(n,N1) Y[n]=-1.2 +.02*n*(1.+.000008*(n-8.)*(n-8));
DO(n,N1) Y[n]=-1. +.04*(n-.5);

//DB sy=6./sinh(.005*N);
//DO(n,N1) Y[n]=sy*sinh(.01*(n-N/2-.5+10));

for(m=-4;m<5;m++) { if(m==0){M(m,-1.1)L(m,6.1)}
                        else {M(m,-1)L(m,6)} }
for(n=-1;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=tai3(z);
                p=Re(c); q=Im(c);
                if(p>-9 && p<9) g[m*N1+n]=p;
                if(q>-9 && q<9) f[m*N1+n]=q;
        }}
p=1;q=.5;
conto(o,g,w,v,X,Y,M,N, ( Re(Zo) ),-q,q); fprintf(o,".1 W 1 .5 1 RGB S\n");
conto(o,f,w,v,X,Y,M,N, ( Im(Zo) ),-q,q); fprintf(o,".1 W .2 1 .5 RGB S\n");
conto(o,f,w,v,X,Y,M,N, (-Im(Zo) ),-q,q); fprintf(o,".1 W .5 1 .2 RGB S\n");

#include"plofu.cin"

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi06a.eps");
        system( "open vladi06a.pdf");//for macintosh
// system( "xpdf fig06a.pdf");//for linux
//getchar(); system("killall Preview");//for macintosh
}

C++ generator of the second picture

fsexp.cin, ado.cin, conto.cin should be loaded in order to compile the code below


#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 "fsexp.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=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("figtai3L.eps","w");ado(o,0,0,82,82);
FILE *o;o=fopen("vladi06b.eps","w");ado(o,82,82);
fprintf(o,"41 11 translate\n 10 10 scale\n");

DO(m,M1) X[m]=-4.+.04*m;
DO(n,N1) Y[n]=-1.+.04*n;

//DB sy=6./sinh(.005*N);
//DO(n,N1) Y[n]=sy*sinh(.01*(n-N/2-.5+10));

for(m=-4;m<5;m++) { if(m==0){M(m,-1.1)L(m,6.1)}
                        else {M(m,-1)L(m,6)} }
for(n=-1;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("run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
        c=log(tai3(z+1.));
        d=tai3(z);
        p=abs(c-d);
        p=-log(p)/log(10.);
        if(p>-999 && p<999) g[m*N1+n]=p;
        }}
fprintf(o,"1 setlinecap 1 setlinejoin\n");

#include"plodi.cin"

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi06b.eps");
        system( "open vladi06b.pdf"); //for macintosh
// system( "xpdf vladi06b.pdf"); //for linux
//getchar(); system("killall Preview");//for macintosh
}

Latex combiner


\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{geometry}
\paperwidth 418px
%\paperheight 134px
\paperheight 190px
\topmargin -108pt
\oddsidemargin -80pt
\pagestyle{empty}
\begin{document}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}

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

\newcommand \taiax {
\put(-3,77){\sx{.5}{$y$}}
\put(-3,69){\sx{.5}{$6$}}
\put(-3,59){\sx{.5}{$5$}}
\put(-3,49){\sx{.5}{$4$}}
\put(-3,39){\sx{.5}{$3$}}
\put(-3,29){\sx{.5}{$2$}}
\put(-3,19){\sx{.5}{$1$}}
\put(-3, 9){\sx{.5}{$0$}}
\put(-5,-1){\sx{.5}{$-\!1$}}
\put( 7 ,-4){\sx{.5}{$-\!3$}}
\put(17 ,-4){\sx{.5}{$-\!2$}}
\put(27 ,-4){\sx{.5}{$-\!1$}}
\put(40 , -4){\sx{.5}{$0$}}
\put(50 , -4){\sx{.5}{$1$}}
\put(60 , -4){\sx{.5}{$2$}}
\put(70 , -4){\sx{.5}{$3$}}
\put(78.8 , -4){\sx{.5}{$x$}}
}
\sx{2.23}{\begin{picture}(100,80)
%\put(0,0){\includegraphics{figtai3}}
\put(0,0){\includegraphics{vladi06a}}
\put(13,55){\sx{.44}{\rot{-77} $ v\!=\!\Im(L)$ \ero }}
\put(23,62.4){\sx{.44}{\rot{-77} $ u\!=\!\Re(L)$ \ero }}
\put(34,65){\sx{.44}{\rot{-76} $ v\!=\!\Im(L)$ \ero }}
\put(45.6,66){\sx{.44}{\rot{-76} $ u\!=\!\Re(L)$ \ero }}
\put(57.6,65){\sx{.44}{\rot{-77} $ v\!=\!\Im(L)$ \ero }}
%\put(33,5){\sx{.4}{\rot{67} $ v\!=\!\Im(L^*)$ \ero }}
\put(42,7){\sx{.4}{\rot{86} $ u\!=\!1$ \ero }}
\put(35,24.6){\sx{.4}{\rot{-31} $ v\!=\!1$ \ero }}
%\put(28,16){\sx{.4}{\rot{37} $ v\!=\!-1$ \ero }}
\taiax
\end{picture}}
\sx{2.23}{\begin{picture}(80,80)
%\put(0,0){\includegraphics{figtai3L}}
\put(0,0){\includegraphics{vladi06b}}
\taiax
\put(26,40){\sx{.58}{$D_{5}\!>\!14$}}
\put(62,15){\sx{.58}{$D_{5}\!<\!1$}}
\end{picture}}
\end{document}

References

  1. ↑ https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
    http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
    http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014.
  2. ↑ 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 6.

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