File:Vladi04.jpg

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Complex map of the truncated Taylor expansion of the natural tetration and the agreements $D_1$ and $D_1$ of this approximation.

Left:

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

$\displaystyle \mathrm{naiv}(z)=\sum_{n=0}^{N-1} c_n z^n$

$\mathrm{tet}(z)=\mathrm{naiv}(z)+O(z^N)$

for $N=50$.

Center:

$\displaystyle D_1= D_{1}(z)=-\lg\left( \frac

$ Right: $\displaystyle D_2=D_{2}(z)=-\lg\left( \frac
$ For $D=D_1$ and $D=D_2$, levels $D=1,2,4,6,8,10,12,14 ~ ~ $ are drawn. Level $D=1$ is drawn with thick line. Symbol "15" indicates the region, where the agreement is better than 14. Usage: this is figure 14.6 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

vladinaiv49.cin, ado.cin, conto.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()
#define I z_type(0.,1.)

#include "vladinaiv49.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=150,M1=M+1;
  int N=301,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("fignaiv.eps","w");ado(o,0,0,62,62);
FILE *o;o=fopen("vladi04a.eps","w");ado(o,62,62);
fprintf(o,"31 31 translate\n 10 10 scale\n");

DO(m,M1) X[m]=-3.+.04*(m-.5);
//DO(n,N1)Y[n]=-6 +.03*(n-.5);

DB sy=2.8/sinh(.005*N);
DO(n,N1) Y[n]=sy*sinh(.01*(n-N/2-.5));

/*
for(m=-20;m<21;m++){ z=z_type(Re(Zo),Im(Zo)*m/20.009); c=FSLOG(z); x=Re(c); y=Im(c); if(m==-20)M(x,y)else L(x,y) }
for(m=20;m>-21;m--){ z=z_type(Re(Zo),Im(Zo)*m/20.009); c=FSLOG(z); x=Re(c)+1;y=Im(c); L(x,y) }
fprintf(o,"1 1 0 RGB F\n");
*/
/*
for(m=-20;m<21;m++){ z=z_type(Re(Zo),Im(Zo)*m/20.008); c=FSLOG(z); x=Re(c);y=Im(c); if(m==-20)M(x,y)else L(x,y) }
for(m=20;m>-21;m--){ z=z_type(Re(Zo),Im(Zo)*m/20.008); c=FSLOG(z); x=Re(c)+1;y=Im(c);if(m==20)M(x,y)else L(x,y) }
fprintf(o,".006 W 0 0 0 RGB S\n");
*/

for(m=-3;m<4;m++) { if(m==0){M(m,-3.2)L(m,3.2)} else {M(m,-3)L(m,3)} }
for(n=-3;n<4;n++) { M( -3,n)L(3,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=naiv49(z);
        p=Re(c); q=Im(c);
        if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
        if(q>-999 && q<999 && fabs(q)> 1.e-8) 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 vladi04a.eps");
        system( "open vladi04a.pdf"); //macintosh
// system( "xpdf vladi04a.pdf"); //linux
//getchar(); system("killall Preview"); //macintosh
}

C++ generator of the second picture


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

//#include "superex.cin"
#include "vladinaiv49.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=150,M1=M+1;
  int N=301,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("fignaivL.eps","w");ado(o,62,62);
FILE *o;o=fopen("vladi04b.eps","w");ado(o,62,62);
fprintf(o,"31 31 translate\n 10 10 scale\n");

DO(m,M1) X[m]=-3.+.04*(m-.5);
//DO(n,N1)Y[n]=-6 +.03*(n-.5);

DB sy=2.8/sinh(.005*N);
DO(n,N1) Y[n]=sy*sinh(.01*(n-N/2-.5));

/*
for(m=-20;m<21;m++){ z=z_type(Re(Zo),Im(Zo)*m/20.009); c=FSLOG(z); x=Re(c); y=Im(c); if(m==-20)M(x,y)else L(x,y) }
for(m=20;m>-21;m--){ z=z_type(Re(Zo),Im(Zo)*m/20.009); c=FSLOG(z); x=Re(c)+1;y=Im(c); L(x,y) }
fprintf(o,"1 1 0 RGB F\n");
*/
/*
for(m=-20;m<21;m++){ z=z_type(Re(Zo),Im(Zo)*m/20.008); c=FSLOG(z); x=Re(c);y=Im(c); if(m==-20)M(x,y)else L(x,y) }
for(m=20;m>-21;m--){ z=z_type(Re(Zo),Im(Zo)*m/20.008); c=FSLOG(z); x=Re(c)+1;y=Im(c);if(m==20)M(x,y)else L(x,y) }
fprintf(o,".006 W 0 0 0 RGB S\n");
*/

for(m=-3;m<4;m++) { if(m==0){M(m,-3.2)L(m,3.2)} else {M(m,-3)L(m,3)} }
for(n=-3;n<4;n++) { M( -3,n)L(3,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(naiv49(z+1.));
        d=naiv49(z);
        c = - log( abs(c-d) / (abs(c)+abs(d)) )/log(10.);
        p=Re(c); //q=Im(c);
        if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
        }}

#include"plodi.cin"
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi04b.eps");
        system( "open vladi04b.pdf");// for macintosh
//getchar(); system("killall Preview");// for macintosh
}

C++ generator of the third picture



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

//#include "superex.cin"
#include "vladinaiv49.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=150,M1=M+1;
  int N=301,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("vladi04c.eps","w");ado(o,62,62);
fprintf(o,"31 31 translate\n 10 10 scale\n");

DO(m,M1) X[m]=-3.+.04*(m-.5);
//DO(n,N1)Y[n]=-6 +.03*(n-.5);

DB sy=2.8/sinh(.005*N);
DO(n,N1) Y[n]=sy*sinh(.01*(n-N/2-.5));

/*
for(m=-20;m<21;m++){ z=z_type(Re(Zo),Im(Zo)*m/20.009); c=FSLOG(z); x=Re(c); y=Im(c); if(m==-20)M(x,y)else L(x,y) }
for(m=20;m>-21;m--){ z=z_type(Re(Zo),Im(Zo)*m/20.009); c=FSLOG(z); x=Re(c)+1;y=Im(c); L(x,y) }
fprintf(o,"1 1 0 RGB F\n");
*/
/*
for(m=-20;m<21;m++){ z=z_type(Re(Zo),Im(Zo)*m/20.008); c=FSLOG(z); x=Re(c);y=Im(c); if(m==-20)M(x,y)else L(x,y) }
for(m=20;m>-21;m--){ z=z_type(Re(Zo),Im(Zo)*m/20.008); c=FSLOG(z); x=Re(c)+1;y=Im(c);if(m==20)M(x,y)else L(x,y) }
fprintf(o,".006 W 0 0 0 RGB S\n");
*/

for(m=-3;m<4;m++) { if(m==0){M(m,-3.2)L(m,3.2)} else {M(m,-3)L(m,3)} }
for(n=-3;n<4;n++) { M( -3,n)L(3,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=exp(naiv49(z-1.));
        d=naiv49(z);
        c = - log( abs(c-d) / (abs(c)+abs(d)) )/log(10.);
        p=Re(c); //q=Im(c);
        if(p>-999 && p<999 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8) g[m*N1+n]=p;
// if(q>-999 && q<999 && fabs(q)> 1.e-8) f[m*N1+n]=q;
        }}

#include"plodi.cin"
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi04c.eps");
        system( "open vladi04c.pdf");// for macintosh
//getchar(); system("killall Preview");// for macintosh
}

Latex combiner


\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{geometry}
\paperwidth 428px
\paperheight 134px
\topmargin -106pt
\oddsidemargin -80pt
\pagestyle{empty}
\begin{document}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}

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

\newcommand \vladiax
{
\put(-3,58.6){\sx{.5}{$y$}}
\put(-3,49){\sx{.5}{$2$}}
\put(-3,39){\sx{.5}{$1$}}
\put(-3,29){\sx{.5}{$0$}}
\put(-7,19){\sx{.5}{$-1$}}
\put(-7, 9){\sx{.5}{$-2$}}
\put( 6 ,-4){\sx{.5}{$-2$}}
\put(17 ,-4){\sx{.5}{$-1$}}
\put(30,-4){\sx{.5}{$0$}}
\put(40, -4){\sx{.5}{$1$}}
\put(50, -4){\sx{.5}{$2$}}
\put(58.4, -4){\sx{.5}{$x$}}
}

%~\sx{2.33}{\begin{picture}(70,60)
~\sx{2.02}{\begin{picture}(70,60)
\put(0,0){\includegraphics{vladi04a}}
\put(25,24){\sx{.4}{\rot{90} $ u\!=\!\Re(L)$ \ero }}
\put(32,51){\sx{.4}{\rot{-61} $ v\!=\!\Im(L)$ \ero }}
\put(27,44){\sx{.4}{\rot{-36} $ v\!=\!1$ \ero }}
\put(26,30){\sx{.4}{\rot{ 0} $ v\!=\!0$ \ero }}
\put(26,15.6){\sx{.4}{\rot{32} $ v\!=\!-1$ \ero }}
\put(35,11){\sx{.4}{\rot{61} $ v\!=\!\Im(L^*)$ \ero }}

\vladiax
\end{picture}}
\sx{2.02}{\begin{picture}(70,60)
\put(0,0){\includegraphics{vladi04b}}
\vladiax
\put(23,29){\sx{.55}{$15$}}
\put(43, 53){\sx{.55}{$D_{1}\!<\!1$}}
\end{picture}}
\sx{2.02}{\begin{picture}(58,60)
\put(0,0){\includegraphics{vladi04c}}
\vladiax
\put(32,29){\sx{.55}{$15$}}
\put(43,53){\sx{.55}{$D_{2}\!<\!1$}}
\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 4.

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