Difference between revisions of "File:Vladi08.jpg"

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{{oq|Vladi08.jpg|Vladi08.jpg ‎(787 × 375 pixels, file size: 117 KB, MIME type: image/jpeg)}}
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Importing image file
  
  +
  +
Fig.14.11 from page 200 of book «[[Superfunctions]]» <ref>
  +
https://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862 <br>
  +
https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3 <br>
  +
https://mizugadro.mydns.jp/BOOK/468.pdf <br>
  +
Dmitrii Kouznetsov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
  +
</ref>, 2020
  +
  +
  +
Agreement of approximations of [[natural tetration]] by elementary functions with the original representation through the Cauchi integral is shown in the left hand side map.
  +
  +
The similar map at the right hand side represents the comparison to the similar representation through the Cauchi integral, but the contour of integration is displaced for 1/2 to the left.
  +
  +
Levels \(D=\mathrm{const}\) are drawn with step 2, but the exception is dome for level \(D=1\), this level is drawn with thick lines.
  +
  +
\(\displaystyle
  +
{\rm fse}(z)=\left\{ \begin{array}{ccrcccr}
  +
{\rm fima}(z) &,& 4.5& \!<\! &\Im(z) \\
  +
{\rm tai}(z) &,& 1.5& \!<\! &\Im(z) &\!\le\!& 4.5 \\
  +
{\rm maclo}(z) &,& -1.5& \le &\Im(z) &\!\le\!& 1.5 \\
  +
{\rm tai}(z^{*})^{*} &,& -4.5& \le &\Im(z) &\!<\!&\!\! -1.5 \\
  +
{\rm fima}(z^{*})^{*} &,& & &\Im(z) &\!<\!&\!\! -4.5
  +
\end{array}
  +
\right.\)
  +
  +
Mnemonics for the name of the approximation: Fast Super Exponent.
  +
  +
The agreement plotted is
  +
  +
\(\displaystyle
  +
D=D_{8}(z)=-\lg\left( \frac
  +
{|\mathrm{fse}(z)-F_{4}(z)|}
  +
{|\mathrm{fse}(z)|+|F_{4}(z)|}
  +
\right)
  +
\)
  +
  +
where \(F_4\) denotes the 4th [[ackermann]], id set, [[natural tetration]] evaluated through the [[Cauchi integral]]
  +
<ref name="analuxp">
  +
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html <br>
  +
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf
  +
D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. [[Mathematics of Computation]], v.78 (2009), 1647-1670.
  +
</ref>.
  +
  +
The same image is used also as Рис.14.11 in the Russian version «[[Суперфункции]]»<ref>
  +
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
  +
http://mizugadro.mydns.jp/BOOK/202.pdf
  +
Д.Кузнецов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014.
  +
</ref>, 2014.
  +
  +
First time published in the [[Vladikavkaz Matehmatical Journal]]
  +
<ref>
  +
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 8.
  +
</ref>, 2010.
  +
  +
The figure suggests that the fast implementation [[fsexp.cin]] with elementary functions happens to
  +
provide a little bit better precision that the initial implementation of [[natural tetration]] through the [[Cauchi integral]] <ref name="analuxp"/> (2009).
  +
  +
==[[C++]] generator of the first picture==
  +
/* [[Fsexp.cin]],
  +
[[ado.cin]],
  +
[[conto.cin]],
  +
[[plodi.cin]]
  +
should be loaded in order to compile the code below */
  +
<pre>
  +
#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 "fsexp.cin"
  +
#include "f4natu.cin"
  +
//z_type FSEXP(z_type z){DB y=Im(z);
  +
  +
z_type fsexp(z_type z){DB y=Im(z);
  +
if(y> 4.5) return fima(z);
  +
if(y> 1.5) return tai3(z);
  +
if(y>-1.5) return maclo(z);
  +
if(y>-4.5) return conj(tai3(conj(z)));
  +
return conj(fima(conj(z)));
  +
}
  +
//#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=100,M1=M+1;
  +
int N=101,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("figcf4small.eps","w");ado(o,0,0,82,82);
  +
FILE *o;o=fopen("vladi08a.eps","w");ado(o,82,82);
  +
fprintf(o,"41 11 translate\n 10 10 scale\n");
  +
  +
DO(m,M1) X[m]=-4.+.08*(m-.5);
  +
DO(n,N1) Y[n]=-1.+.08*(n-.5);
  +
  +
for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
  +
for(n=0;n<7;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=fsexp(z);
  +
d=F4natu(z);
  +
p=abs(c-d);///(abs(c)+abs(d));
  +
p=-log(p)/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"
  +
//M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash S\n");
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
// system( "gv fig08a.eps");
  +
system("epstopdf vladi08a.eps");
  +
system( "open vladi08a.pdf"); // for linux
  +
// getchar(); system("killall Preview");// for macintosh
  +
}
  +
  +
</pre>
  +
  +
==[[C++]] generator of the second picture==
  +
  +
/* [[vladif5c.cin]] also should be also loaded */
  +
<pre>
  +
#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 "fsexp.cin"
  +
//#include "f4natu.cin"
  +
#include "vladif5c.cin"
  +
//z_type FSEXP(z_type z){DB y=Im(z);
  +
  +
z_type fsexp(z_type z){DB y=Im(z);
  +
if(y> 4.5) return fima(z);
  +
if(y> 1.5) return tai3(z);
  +
if(y>-1.5) return maclo(z);
  +
if(y>-4.5) return conj(tai3(conj(z)));
  +
return conj(fima(conj(z)));
  +
}
  +
//#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=100,M1=M+1;
  +
int N=101,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("figcf5small.eps","w");ado(o,0,0,82,82);
  +
FILE *o;o=fopen("vladi08b.eps","w");ado(o,82,82);
  +
fprintf(o,"41 11 translate\n 10 10 scale\n");
  +
  +
DO(m,M1)X[m]=-4.+.08*(m-.5);
  +
DO(n,N1)Y[n]=-1.+.08*(n-.5);
  +
  +
for(m=-3;m<4;m++) { if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)} }
  +
for(n= 0;n<7;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=fsexp(z);
  +
d=F5(z);
  +
p=abs(c-d);///(abs(c)+abs(d));
  +
p=-log(p)/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"
  +
//M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash S\n");
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf vladi08b.eps");
  +
system( "open vladi08b.pdf");//linux
  +
//getchar(); system("killall Preview");
  +
}
  +
</pre>
  +
==[[Latex]] combiner==
  +
<pre>
  +
\documentclass[12pt]{article}
  +
\usepackage{graphicx}
  +
\usepackage{rotating}
  +
\usepackage{geometry}
  +
\paperwidth 378px
  +
%\paperheight 134px
  +
\paperheight 180px
  +
\topmargin -104pt
  +
\oddsidemargin -94pt
  +
\pagestyle{empty}
  +
\begin{document}
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \sx {\scalebox}
  +
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
  +
\sx{2.2}{\begin{picture}(90,80)
  +
%\put(0,0){\includegraphics{figsexpF4}}
  +
%\put(0,0){\includegraphics{figcf4small}}
  +
\put(0,0){\includegraphics{vladi08a}}
  +
\put(5,68){\sx{.45}{$\Im(z)$}}
  +
\put(7,59){\sx{.5}{$5$}}
  +
\put(7,49){\sx{.5}{$4$}}
  +
\put(7,39){\sx{.5}{$3$}}
  +
\put(7,29){\sx{.5}{$2$}}
  +
\put(7,19){\sx{.5}{$1$}}
  +
\put(7, 9){\sx{.5}{$0$}}
  +
\put(70 , 6){\sx{.45}{$\Re(z)$}}
  +
\put(60 , 6){\sx{.5}{$2$}}
  +
\put(40 , 6){\sx{.5}{$0$}}
  +
\put(17 ,6){\sx{.5}{$-\!2$}}
  +
\put(25,62){\sx{.5}{$D\!>\!14$}}
  +
\put(55,60){\sx{.5}{$12\!<\!D\!<\!14$}}
  +
\put(68,20){\sx{.5}{$D\!<\!1$}}
  +
\end{picture}}
  +
\sx{2.2}{\begin{picture}(80,80)
  +
%\put(0,0){\includegraphics{figsexpF5}}
  +
%\put(0,0){\includegraphics{figcf5small}}
  +
\put(0,0){\includegraphics{vladi08b}}
  +
\put(5,68){\sx{.45}{$\Im(z)$}}
  +
\put(7,59){\sx{.5}{$5$}}
  +
\put(7,49){\sx{.5}{$4$}}
  +
\put(7,39){\sx{.5}{$3$}}
  +
\put(7,29){\sx{.5}{$2$}}
  +
\put(7,19){\sx{.5}{$1$}}
  +
\put(7, 9){\sx{.5}{$0$}}
  +
\put(70 , 6){\sx{.45}{$\Re(z)$}}
  +
\put(60 , 6){\sx{.5}{$2$}}
  +
\put(40 , 6){\sx{.5}{$0$}}
  +
\put(17 ,6){\sx{.5}{$-\!2$}}
  +
\put(31,56){\sx{.5}{$D\!>\!14$}}
  +
\put(68,20){\sx{.5}{$D\!<\!1$}}
  +
\end{picture}}
  +
\end{document}
  +
</pre>
  +
==Refereces==
  +
{{ref}}
  +
  +
{{fer}}
  +
==Keywords==
  +
«[[Agreement]]»,
  +
«[[Approximation]]»,
  +
«[[Natural tetration]]»,
  +
«[[Superfunction]]»,
  +
«[[Superfunctions]]»,
  +
«[[Tetration]]»,
  +
  +
[[Category:Agreement]]
  +
[[Category:Approximation]]
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:C++]]
  +
[[Category:Latex]]
  +
[[Category:Natural tetration]]
  +
[[Category:Superfunction]]
  +
[[Category:Superfunctions]]
  +
[[Category:Test]]
  +
[[Category:Tetration]]

Latest revision as of 20:59, 17 December 2025


Fig.14.11 from page 200 of book «Superfunctions» [1], 2020


Agreement of approximations of natural tetration by elementary functions with the original representation through the Cauchi integral is shown in the left hand side map.

The similar map at the right hand side represents the comparison to the similar representation through the Cauchi integral, but the contour of integration is displaced for 1/2 to the left.

Levels \(D=\mathrm{const}\) are drawn with step 2, but the exception is dome for level \(D=1\), this level is drawn with thick lines.

\(\displaystyle {\rm fse}(z)=\left\{ \begin{array}{ccrcccr} {\rm fima}(z) &,& 4.5& \!<\! &\Im(z) \\ {\rm tai}(z) &,& 1.5& \!<\! &\Im(z) &\!\le\!& 4.5 \\ {\rm maclo}(z) &,& -1.5& \le &\Im(z) &\!\le\!& 1.5 \\ {\rm tai}(z^{*})^{*} &,& -4.5& \le &\Im(z) &\!<\!&\!\! -1.5 \\ {\rm fima}(z^{*})^{*} &,& & &\Im(z) &\!<\!&\!\! -4.5 \end{array} \right.\)

Mnemonics for the name of the approximation: Fast Super Exponent.

The agreement plotted is

\(\displaystyle D=D_{8}(z)=-\lg\left( \frac {|\mathrm{fse}(z)-F_{4}(z)|} {|\mathrm{fse}(z)|+|F_{4}(z)|} \right) \)

where \(F_4\) denotes the 4th ackermann, id set, natural tetration evaluated through the Cauchi integral [2].

The same image is used also as Рис.14.11 in the Russian version «Суперфункции»[3], 2014.

First time published in the Vladikavkaz Matehmatical Journal [4], 2010.

The figure suggests that the fast implementation fsexp.cin with elementary functions happens to provide a little bit better precision that the initial implementation of natural tetration through the Cauchi integral [2] (2009).

C++ generator of the first picture

/* Fsexp.cin, ado.cin, conto.cin, plodi.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 "fsexp.cin"
#include "f4natu.cin"
//z_type FSEXP(z_type z){DB y=Im(z);

z_type fsexp(z_type z){DB y=Im(z);
 if(y> 4.5) return fima(z);
 if(y> 1.5) return tai3(z);
 if(y>-1.5) return maclo(z);
 if(y>-4.5) return conj(tai3(conj(z)));
            return conj(fima(conj(z)));
}
//#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=100,M1=M+1;
  int N=101,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("figcf4small.eps","w");ado(o,0,0,82,82);
FILE *o;o=fopen("vladi08a.eps","w");ado(o,82,82);
fprintf(o,"41 11 translate\n 10 10 scale\n");

DO(m,M1) X[m]=-4.+.08*(m-.5);
DO(n,N1) Y[n]=-1.+.08*(n-.5);

for(m=-3;m<4;m++) {     if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)}                 }
for(n=0;n<7;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=fsexp(z);
        d=F4natu(z);
        p=abs(c-d);///(abs(c)+abs(d)); 
        p=-log(p)/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"
//M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash  S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
//      system(    "gv fig08a.eps");
        system("epstopdf vladi08a.eps");
        system(    "open vladi08a.pdf"); // for linux
//      getchar(); system("killall Preview");// for macintosh
}

C++ generator of the second picture

/* vladif5c.cin also should be also loaded */ 

#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 "fsexp.cin"
//#include "f4natu.cin"
#include "vladif5c.cin"
//z_type FSEXP(z_type z){DB y=Im(z);

z_type fsexp(z_type z){DB y=Im(z);
 if(y> 4.5) return fima(z);
 if(y> 1.5) return tai3(z);
 if(y>-1.5) return maclo(z);
 if(y>-4.5) return conj(tai3(conj(z)));
            return conj(fima(conj(z)));
}
//#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=100,M1=M+1;
  int N=101,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("figcf5small.eps","w");ado(o,0,0,82,82);
FILE *o;o=fopen("vladi08b.eps","w");ado(o,82,82);
fprintf(o,"41 11 translate\n 10 10 scale\n");

DO(m,M1)X[m]=-4.+.08*(m-.5);
DO(n,N1)Y[n]=-1.+.08*(n-.5);

for(m=-3;m<4;m++) {     if(m==0){M(m,-0.1)L(m,6.1)} else {M(m,0)L(m,6)}                 }
for(n= 0;n<7;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=fsexp(z);
        d=F5(z);
        p=abs(c-d);///(abs(c)+abs(d));
        p=-log(p)/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"
//M(-2,0)L(-3.1,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash  S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi08b.eps");
        system(    "open vladi08b.pdf");//linux
//getchar(); system("killall Preview");
}

Latex combiner

\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{geometry}
\paperwidth 378px
%\paperheight 134px 
\paperheight 180px 
\topmargin -104pt
\oddsidemargin -94pt
\pagestyle{empty}
\begin{document}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}

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

\sx{2.2}{\begin{picture}(90,80)
%\put(0,0){\includegraphics{figsexpF4}}
%\put(0,0){\includegraphics{figcf4small}}
\put(0,0){\includegraphics{vladi08a}}
\put(5,68){\sx{.45}{$\Im(z)$}}
\put(7,59){\sx{.5}{$5$}}
\put(7,49){\sx{.5}{$4$}}
\put(7,39){\sx{.5}{$3$}}
\put(7,29){\sx{.5}{$2$}}
\put(7,19){\sx{.5}{$1$}}
\put(7, 9){\sx{.5}{$0$}}
\put(70 , 6){\sx{.45}{$\Re(z)$}}
\put(60 , 6){\sx{.5}{$2$}}
\put(40 , 6){\sx{.5}{$0$}}
\put(17  ,6){\sx{.5}{$-\!2$}}
\put(25,62){\sx{.5}{$D\!>\!14$}}
\put(55,60){\sx{.5}{$12\!<\!D\!<\!14$}}
\put(68,20){\sx{.5}{$D\!<\!1$}}
\end{picture}}
\sx{2.2}{\begin{picture}(80,80)
%\put(0,0){\includegraphics{figsexpF5}}
%\put(0,0){\includegraphics{figcf5small}}
\put(0,0){\includegraphics{vladi08b}}
\put(5,68){\sx{.45}{$\Im(z)$}}
\put(7,59){\sx{.5}{$5$}}
\put(7,49){\sx{.5}{$4$}}
\put(7,39){\sx{.5}{$3$}}
\put(7,29){\sx{.5}{$2$}}
\put(7,19){\sx{.5}{$1$}}
\put(7, 9){\sx{.5}{$0$}}
\put(70 , 6){\sx{.45}{$\Re(z)$}}
\put(60 , 6){\sx{.5}{$2$}}
\put(40 , 6){\sx{.5}{$0$}}
\put(17  ,6){\sx{.5}{$-\!2$}}
\put(31,56){\sx{.5}{$D\!>\!14$}}
\put(68,20){\sx{.5}{$D\!<\!1$}}
\end{picture}}
\end{document}

Refereces

  1. ↑ https://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862
    https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3
    https://mizugadro.mydns.jp/BOOK/468.pdf
    Dmitrii Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.
  2. ↑ 2.0 2.1 http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
    http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.
  3. ↑ https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
    http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014.
  4. ↑ 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 8.

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