Difference between revisions of "File:Vladi05.jpg"

Revision as of 20:42, 10 October 2025

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.

Use

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

It appears also as figure 14.7 in the Russian version «Суперфункции», 2014 ^[2].

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

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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[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 $D$.
+[[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]]
 <b>Left:</b>
-$u\!+\!\mathrm i v = \mathrm{maclo}(x+\mathrm i y)$
+\(u\!+\!\mathrm i v = \mathrm{maclo}(x+\mathrm i y)\)
-$\displaystyle
+\(\displaystyle
-\mathrm{maclo}(z)= \ln(z\!+\!2)+\sum_{n=0}^{N-1} s_n z^n$
+\mathrm{maclo}(z)= \ln(z\!+\!2)+\sum_{n=0}^{N-1} s_n z^n\)
-$\mathrm{tet}(z)=\mathrm{maclo}(z)+O(z^N)$
+\(\mathrm{tet}(z)=\mathrm{maclo}(z)+O(z^N)\)
-for $N=101$.
+for \(N=101\).
 <b>Center:</b>
-$\displaystyle D_1=
+\(\displaystyle D_1=
 D_{1}(z)=-\lg\left( \frac
-{|\ln(\mathrm{maclo}(z\!+\!1)-\mathrm{naiv}(z)|}
+{|\ln(\mathrm{maclo}(z\!+\!1)-\mathrm{maclo}(z)|}
-{|\ln(\mathrm{maclo}(z\!+\!1)|+|\mathrm{naiv}(z)|} \right)
+{|\ln(\mathrm{maclo}(z\!+\!1)|+|\mathrm{maclo}(z)|} \right)
+\)
-$
 <b>Right:</b>
-$\displaystyle
+\(\displaystyle
 D_2=D_{2}(z)=-\lg\left( \frac
 {|\exp(\mathrm{maclo}(z\!-\!1)-\mathrm{maclo}(z)|}
 {|\exp(\mathrm{maclo}(z\!-\!1)|+|\mathrm{maclo}(z)|} \right)
+\)
-$
-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.
+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.
+==Use==
-Usage: this is figure 14.7 of the book [[Суперфункции]] (2014, In Russian)
+This picture appears as 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 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://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>
+</ref>.
-; the English version is in preparation in 2015.
 First time published in the [[Vladikavkaz Matehmatical Journal]]
@@ Line 45: / Line 55: @@
 D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.
 Figure 5.
-</ref>.
+</ref>, 2010.
 ==[[C++]] generator of the first picture==
@@ Line 52: / Line 62: @@
 [[conto.cin]]
 should be loaded in order to compile the code below
+<pre>
-<poem><nomathjax><nowiki>
 #include <stdio.h>
 #include <stdlib.h>
@@ Line 125: / Line 134: @@
 }
+</pre>
-</nowiki></nomathjax></poem>
 ==[[C++]] generator of the second picture==
+<pre>
-<poem><nomathjax><nowiki>
 #include <stdio.h>
 #include <stdlib.h>
@@ Line 194: / Line 203: @@
 }
+</pre>>
-</nowiki></nomathjax></poem>
 ==[[C++]] generator of the right picture==
+<pre>
-<poem><nomathjax><nowiki>
 #include <stdio.h>
 #include <stdlib.h>
@@ Line 261: / Line 270: @@
 }
+</pre>
-</nowiki></nomathjax></poem>
 ==[[Latex]] combiner==
+<pre>
-<poem><nomathjax><nowiki>
 \documentclass[12pt]{article}
 \usepackage{graphicx}
@@ Line 323: / Line 332: @@
 \end{document}
+</pre>
-</nowiki></nomathjax></poem>
+==References==
+{{ref}}
+{{fer}}
+==Keywords==
+«[[Agreement]]»,
+«[[fsexp.cin]]»,
+«[[Natural tetration]]»,
+«[[Plodi.cin]]»,
+«[[Superfunctions]]»,
+«[[Tetration]]»,
+«[[Суперфункции]]»,
+[[Category:Agreement]]
-==References==
+[[Category:Approximation]]
-<references/>
 [[Category:Book]]
-[[Category:Agreement]]
-[[Category:Complex map]]
 [[Category:BookMap]]
-[[Category:Tetration]]
 [[Category:C++]]
+[[Category:Complex map]]
 [[Category:Latex]]
+[[Category:Superfunctions]]
+[[Category:Tetration]]