Difference between revisions of "File:Vladi05.jpg"

Latest revision as of 08:04, 16 December 2025

Summary

Fig.14.8 at page 195 of book «Superfunctions», 2020 ^[1].

It appears also as Рис.14.7 at page 193 the Russian version «Суперфункции», 2014 ^[2].

First time published in the Vladikavkaz Matehmatical Journal ^[3], 2010.

The figure shows the Complex map of approximation maclo of the natural tetration, left, and maps of the agreements \(D\). Approximation maclo is used in the C++ implementation fsexp.cin of Natural tetration

Left:

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

\(\displaystyle \mathrm{maclo}(z)= \ln(z\!+\!2)+\sum_{n=0}^{N-1} s_n z^n\)

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

for \(N=101\).

Center:

\(\displaystyle D_1= D_{1}(z)=-\lg\left( \frac {|\ln(\mathrm{maclo}(z\!+\!1)-\mathrm{maclo}(z)|} {|\ln(\mathrm{maclo}(z\!+\!1)|+|\mathrm{maclo}(z)|} \right) \)

Right:

\(\displaystyle D_2=D_{2}(z)=-\lg\left( \frac {|\exp(\mathrm{maclo}(z\!-\!1)-\mathrm{maclo}(z)|} {|\exp(\mathrm{maclo}(z\!-\!1)|+|\mathrm{maclo}(z)|} \right) \)

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.

Coefficients of the expansion had been evaluated using the primary approximations through the Cauchi integral.

The agreements show, that the approximation "maclo" can serve to plot maps of the natural tetration \(\mathrm{tet}(z)\) for \(\Im(z)<2.5\); and for \(\Im(z)\le 2\), the precision of the evaluation is even better than that by the primary approximation through the Cauchi integral.

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 "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=250,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("figmaclo.eps","w");ado(o,0,0,62,62);
FILE *o;o=fopen("vladi05a.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=maclo(z);
        p=Re(c); q=Im(c);
        if(p>-9 && p<9 && fabs(p)> 1.e-8 && fabs(p-1.)>1.e-8)   g[m*N1+n]=p;
        if(q>-9 && q<9 && 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"

M(-2,0)L(-3.03,0) fprintf(o,".07 W 1 1 1 RGB S\n");
DO(m,6){ M(-2-.25*m,0)L(-2-.25*(m+.5),0)} fprintf(o,".08 W 0 0 0 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf vladi05a.eps");
        system(    "open vladi05a.pdf");//for macintosh
//      system(    "xpdf fig05a.pdf");//for linux
//getchar(); system("killall Preview");//for macintosh
}

C++ generator of the second picture

#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 "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("figmacloL.eps","w");ado(o,0,0,62,62);
FILE *o;o=fopen("vladi05b.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(maclo(z+1.));
        d=maclo(z);
        p= abs(c-d)/(abs(c)+abs(d));
        p=-log(p)/log(10.);
        // p=Re(c); q=Im(c);
        if(p>-99 && p<99 && 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 vladi05b.eps");
        system(    "open vladi05b.eps"); // for macintosh
//getchar(); system("killall Preview"); //for macintosh
}

>

C++ generator of the right picture

#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 "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("vladi05c.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(maclo(z-1.));
        d=maclo(z);
        p= abs(c-d)/(abs(c)+abs(d));
        p=-log(p)/log(10.);
        // p=Re(c); q=Im(c);
        if(p>-99 && p<99 && 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 vladi05c.eps");
        system(    "open vladi05c.eps"); // 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{vladi05a}}
\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(32.6,26){\sx{.4}{\rot{90} $ u\!=\!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{vladi05b}}
\vladiax
\put(23,29){\sx{.55}{$15$}}
\put(43, 55){\sx{.55}{$D_{1}\!<\!1$}}
\end{picture}}
\sx{2.02}{\begin{picture}(58,60)
\put(0,0){\includegraphics{vladi05c}}
\vladiax
\put(32,29){\sx{.55}{$15$}}
\put(44,56){\sx{.55}{$D_{2}\!<\!1$}}
\end{picture}}

\end{document}

References

↑ 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.
↑ 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.
↑ 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 5.

Keywords

«Agreement», «fsexp.cin», «Natural tetration», «Plodi.cin», «Superfunctions», «Tetration», «Суперфункции»,

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Vertical resolution	300 dpi

[1] ttps://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] ttps://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.

[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 5.

[1]

[2]

[3]

@@ Line 1: / Line 1: @@
 {{oq|Vladi05.jpg|Original file ‎(1,783 × 558 pixels, file size: 342 KB, MIME type: image/jpeg)}}
-[[Complex map]] of approximation [[maclo]] of the natural [[tetration]], left, and maps of the agreements \(D\).
+==Summary==
+Fig.14.8 at page 195 of book «[[Superfunctions]]», 2020 <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>.
+It appears also as Рис.14.7 at page 193 the Russian version «[[Суперфункции]]», 2014
+<ref>
+https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
+<!--http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf <br> !-->
+http://mizugadro.mydns.jp/BOOK/202.pdf
+Д.Кузнецов. Суперфункции. [[Lambert Academic Publishing]], 2014.
+</ref>.
+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 5.
+</ref>, 2010.
+The figure shows the
+[[Complex map]] of approximation [[maclo]] of the [[natural tetration]], left, and maps of the agreements \(D\).
 Approximation [[maclo]] is used in the [[C++]] implementation [[fsexp.cin]] of [[Natural tetration]]
@@ Line 33: / Line 59: @@
 Symbol "15" indicates the region, where the agreement is better than 14.
+Coefficients of the expansion had been evaluated using the primary approximations through the [[Cauchi integral]].
-==Use==
+The agreements show, that the approximation "maclo" can serve to plot maps of the [[natural tetration]]
-This picture appears as Fig.14.8 at page 195 of book «[[Superfunctions]]», 2020 <ref>
+\(\mathrm{tet}(z)\) for \(\Im(z)<2.5\); and for \(\Im(z)\le 2\), the precision of the evaluation
-https://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862 <br>
+is even better than that by the primary approximation through the [[Cauchi integral]].
-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>.
-It appears also as figure 14.7 in the Russian version «[[Суперфункции]]», 2014
-<ref>
-https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
-<!--http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf <br> !-->
-http://mizugadro.mydns.jp/BOOK/202.pdf
-Д.Кузнецов. Суперфункции. [[Lambert Academic Publishing]], 2014.
-</ref>.
-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 5.
-</ref>, 2010.
 ==[[C++]] generator of the first picture==