Difference between revisions of "File:Tetsheldonmap03.png"

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  +
{{oq|Tetsheldonmap03.png|Original file ‎(2,549 × 702 pixels, file size: 982 KB, MIME type: image/png)}}
−
[[Complex map]] of [[Tetration to Sheldon base]]
  
   
  +
Top picture of Figure 18.3 at page 250 of book «[[Superfunctions]]»
−
Copy from the duplicate of TORI
  
  +
<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>
  +
D.Kouznetov. [[Superfunctions]]. [[Lambert Academic Publishing]], 2020.
  +
</ref>:
   
  +
[[Complex map]] of [[tetration to Sheldon base]]
−
[[Category:Tetration]]
  
  +
\(b=s=1.52598338517+0.0178411853321 i \).
−
[[Category:Sheldon base]]
  
  +
  +
Function \(f=\mathrm{tet}_s(x+\mathrm i y)\) is shown in the \(x,y\) plane
  +
with levels \(u=\Re(f)=\mathrm{const}\)
  +
and levels \(v=\Im(f)=\mathrm{const}\);
  +
thick lines correspond ot the integer values.
  +
  +
This map appears also at page 257 as "Рис.18.3" of the Russian version «[[Суперфункции]]»
  +
of the same book <ref>
  +
https://mizugadro.mydns.jp/BOOK/202.pdf <br>
  +
Дмитрий Кузнецов. [[Суперфункции]]. [[Lambert Academic Publishing]], 2014
  +
</ref>
  +
  +
==[[C++]] generator of map==
  +
/* Files
  +
[[ado.cin]],
  +
[[conto.cin]],
  +
[[filog.cin]],
  +
[[GLxw2048.inc]],
  +
[[TetSheldonIma.inc]]
  +
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++)
  +
//using namespace std;
  +
#include <complex>
  +
typedef std::complex<double> z_type;
  +
// typedef complex<double> z_type;
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
#include "conto.cin"
  +
#include "filog.cin"
  +
z_type b=z_type( 1.5259833851700000, 0.0178411853321000);
  +
z_type a=log(b);
  +
z_type Zo=Filog(a);
  +
z_type Zc=conj(Filog(conj(a)));
  +
DB A=32.;
  +
z_type tetb(z_type z){ int k; DB t; z_type c, cu,cd;
  +
#include "GLxw2048.inc"
  +
int K=2048;
  +
//#include "ima6.inc"
  +
#include "TetSheldonIma.inc"
  +
z_type E[2048],G[2048];
  +
DO(k,K){c=F[k]; E[k]=log(c)/a; G[k]=exp(a*c);}
  +
c=0.;
  +
z+=z_type(0.1196573712872846, 0.1299776198056910);
  +
DO(k,K){t=A*GLx[k];c+=GLw[k]*(G[k]/(z_type( 1.,t)-z)-E[k]/(z_type(-1.,t)-z));}
  +
cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) );
  +
cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) );
  +
c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd;
  +
return c;}
  +
  +
int main(){ int j,k,m,m1,n; DB x,y, p,q, t; z_type z,c,d;
  +
//int M=161,M1=M+1;
  +
int M=601,M1=M+1;
  +
int N=401,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("tetsheldonmap.eps","w");ado(o,602,202);
  +
fprintf(o,"301 101 translate\n 10 10 scale\n");
  +
DO(m,M1)X[m]=-30.+.1*(m);
  +
DO(n,200)Y[n]=-10.+.05*n;
  +
Y[200]=-.01;
  +
Y[201]= .01;
  +
for(n=202;n<N1;n++) Y[n]=-10.+.05*(n-1.);
  +
for(m=-30;m<31;m++){if(m==0){M(m,-10.2)L(m,10.2)} else{M(m,-10)L(m,10)}}
  +
for(n=-10;n<11;n++){ M( -30,n)L(30,n)}
  +
fprintf(o,".008 W 0 0 0 RGB S\n");
  +
DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
  +
DO(n,N1){y=Y[n];
  +
for(m=295;m<305;m++)
  +
{x=X[m]; //printf("%5.2f\n",x);
  +
z=z_type(x,y);
  +
c=tetb(z);
  +
p=Re(c);q=Im(c);
  +
if(p>-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
  +
d=c;
  +
for(k=1;k<31;k++)
  +
{ m1=m+k*10; if(m1>M) break;
  +
d=exp(a*d);
  +
p=Re(d);q=Im(d);
  +
if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
  +
}
  +
d=c;
  +
for(k=1;k<31;k++)
  +
{ m1=m-k*10; if(m1<0) break;
  +
d=log(d)/a;
  +
p=Re(d);q=Im(d);
  +
if(p>-99. && p<99. && q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
  +
}
  +
}}
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1;q=.5;
  +
for(m=-10;m<10;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".02 W 0 .6 0 RGB S\n");
  +
for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".02 W .9 0 0 RGB S\n");
  +
for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".02 W 0 0 .9 RGB S\n");
  +
for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".08 W .9 0 0 RGB S\n");
  +
for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 .9 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".08 W .6 0 .6 RGB S\n");
  +
for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 0 RGB S\n");
  +
// y= 0; for(m=0;m<260;m+=6) {x=-2.-.1*m; M(x,y) L(x-.1,y)}
  +
// fprintf(o,".07 W 1 .5 0 RGB S\n");
  +
// y= 0; for(m=3;m<260;m+=6) {x=-2-.1*m; M(x,y) L(x-.1,y)}
  +
// fprintf(o,".07 W 0 .5 1 RGB S\n");
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf tetsheldonmap.eps");
  +
system( "open tetsheldonmap.pdf");
  +
getchar(); system("killall Preview");
  +
}
  +
</pre>
  +
==[[Latex]] generator of labels==
  +
<pre>
  +
\documentclass[12pt]{article}
  +
\paperwidth 618pt
  +
\paperheight 214pt
  +
\textwidth 1060pt
  +
\textheight 500pt
  +
\topmargin -108pt
  +
\oddsidemargin -70pt
  +
\pagestyle{empty}
  +
\usepackage[usenames]{color}
  +
\usepackage[utf8x]{inputenc}
  +
\usepackage{hyperref}
  +
\usepackage{graphicx}
  +
\usepackage{rotating}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \rme {\mathrm{e}}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\begin{document}
  +
\parindent 0pt
  +
\begin{picture}(202,211)
  +
\put(10,10){\ing{tetsheldonmap}}
  +
\put(2,206){$y$}
  +
\put(2,188){$8$}
  +
\put(2,168){$6$}
  +
\put(2,148){$4$}
  +
\put(2,128){$2$}
  +
\put(2,108){$0$}
  +
\put(-6,88){$-2$}
  +
\put(-6,68){$-4$}
  +
\put(-6,48){$-6$}
  +
\put(-6,28){$-8$}
  +
%\put(0,8){-10}\put(261,0){$-4$}
  +
\put(-1,0){$-30$}
  +
\put( 49,0){$-25$}
  +
\put( 99,0){$-20$}
  +
\put(149,0){$-15$}
  +
\put(199,0){$-10$}
  +
%\put(222,0){$-8$}
  +
%\put(242,0){$-6$}
  +
\put(252,0){$-5$}
  +
%\put(262,0){$-4$}
  +
%\put(282,0){$-2$}
  +
\put(309,0){$0$}
  +
\put(329,0){$2$}
  +
\put(349,0){$4$}
  +
\put(369,0){$6$}
  +
\put(389,0){$8$}
  +
\put(407,0){$10$}
  +
\put(457,0){$15$}
  +
\put(507,0){$20$}
  +
\put(557,0){$25$}
  +
\put(607,1){$x$}
  +
\multiput(24,180)(118,-1){5}{$v\!=\!1.2$}
  +
\multiput(70,180)(118,-1){5}{$u\!=\!2$}
  +
\multiput(112,158)(118,-1){5}{\rot{-50}$v\!=\!1$\ero}
  +
\multiput(102,142)(118,-1){5}{\rot{-35}$v\!=\!0.8$\ero}
  +
\put(336,107){\rot{-11}$v\!=\!0$\ero}
  +
\multiput(206, 92)(119,-22){4}{$v\!=\!-1$}
  +
\multiput(70, 84)(119,-22){4}{$u\!=\!1.2$}
  +
\multiput(110, 76)(119,-22){4}{$v\!=\!-1.4$}
  +
\put(470,200){$u+\mathrm i v \approx 2.0565+1.1445 \,\mathrm i$}
  +
\put(30,20){$u+\mathrm i v \approx 2.2284-1.3508 \,\mathrm i$}
  +
\end{picture}
  +
\end{document}
  +
</pre>
  +
  +
==References==
  +
{{ref}}
  +
  +
{{fer}}
  +
  +
==Keywords==
  +
«[[Abelfunction]]»,
  +
«[[Superfunction]]»,
  +
«[[Superfunctions]]»,
  +
«[[Tetration to Sheldon base]]»,
  +
«[[Tetration]]»,
  +
  +
«[[ado.cin]]»,
  +
«[[conto.cin]]»,
  +
«[[filog.cin]]»,
  +
«[[GLxw2048.inc]]»,
  +
«[[TetSheldonIma.inc]]»,
  +
  +
«[[Суперфункции]]»,
  +
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:C++]]
 
[[Category:Complex map]]
  
[[Category:Complex map]]
  +
[[Category:Lambert Academic Publishing]]
  +
[[Category:Latex]]
  +
[[Category:Tetration]]
  +
[[Category:Tetration to Sheldon base]]
  +
[[Category:Sheldon Levenstein]]
  +
[[Category:Superfunction]]
  +
[[Category:Superfunctions]]

Latest revision as of 17:54, 9 December 2025


Top picture of Figure 18.3 at page 250 of book «Superfunctions» [1]:

Complex map of tetration to Sheldon base \(b=s=1.52598338517+0.0178411853321 i \).

Function \(f=\mathrm{tet}_s(x+\mathrm i y)\) is shown in the \(x,y\) plane with levels \(u=\Re(f)=\mathrm{const}\) and levels \(v=\Im(f)=\mathrm{const}\); thick lines correspond ot the integer values.

This map appears also at page 257 as "Рис.18.3" of the Russian version «Суперфункции» of the same book [2]

C++ generator of map

/* Files ado.cin, conto.cin, filog.cin, GLxw2048.inc, TetSheldonIma.inc 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;
//  typedef complex<double> z_type;
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)
 #include "conto.cin"
 #include "filog.cin"
z_type b=z_type( 1.5259833851700000, 0.0178411853321000);
z_type a=log(b);
z_type Zo=Filog(a);
z_type Zc=conj(Filog(conj(a)));
DB A=32.;
z_type tetb(z_type z){ int k; DB t; z_type c, cu,cd;
#include "GLxw2048.inc"
int K=2048;
//#include "ima6.inc"
#include "TetSheldonIma.inc"
z_type E[2048],G[2048];
DO(k,K){c=F[k]; E[k]=log(c)/a; G[k]=exp(a*c);}
c=0.;
z+=z_type(0.1196573712872846, 0.1299776198056910);
DO(k,K){t=A*GLx[k];c+=GLw[k]*(G[k]/(z_type( 1.,t)-z)-E[k]/(z_type(-1.,t)-z));}  
 cu=.5-I/(2.*M_PI)*log( (z_type(1.,-A)+z)/(z_type(1., A)-z) );
 cd=.5-I/(2.*M_PI)*log( (z_type(1.,-A)-z)/(z_type(1., A)+z) );
 c=c*(A/(2.*M_PI)) +Zo*cu+Zc*cd;
 return c;}

int main(){ int j,k,m,m1,n; DB x,y, p,q, t; z_type z,c,d;
 //int M=161,M1=M+1;
 int M=601,M1=M+1;
 int N=401,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("tetsheldonmap.eps","w");ado(o,602,202);
 fprintf(o,"301 101 translate\n 10 10 scale\n");
 DO(m,M1)X[m]=-30.+.1*(m);
 DO(n,200)Y[n]=-10.+.05*n;
         Y[200]=-.01;
         Y[201]= .01;
 for(n=202;n<N1;n++) Y[n]=-10.+.05*(n-1.);
 for(m=-30;m<31;m++){if(m==0){M(m,-10.2)L(m,10.2)} else{M(m,-10)L(m,10)}}
 for(n=-10;n<11;n++){   M(  -30,n)L(30,n)}
 fprintf(o,".008 W 0 0 0 RGB S\n");
 DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
 DO(n,N1){y=Y[n]; 
          for(m=295;m<305;m++)
          {x=X[m]; //printf("%5.2f\n",x);
           z=z_type(x,y);       
           c=tetb(z); 
           p=Re(c);q=Im(c);     
           if(p>-99. && p<99. &&  q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;}
           d=c; 
           for(k=1;k<31;k++)
                { m1=m+k*10; if(m1>M) break; 
                d=exp(a*d); 
                p=Re(d);q=Im(d);        
                if(p>-99. && p<99. &&  q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
                }
           d=c; 
           for(k=1;k<31;k++)
                { m1=m-k*10; if(m1<0) break; 
                d=log(d)/a; 
                p=Re(d);q=Im(d);        
                if(p>-99. && p<99. &&  q>-99. && q<99. ){ g[m1*N1+n]=p;f[m1*N1+n]=q;}
                }
        }}
 fprintf(o,"1 setlinejoin 2 setlinecap\n");  p=1;q=.5;
 for(m=-10;m<10;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".02 W 0 .6 0 RGB S\n");
 for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".02 W .9 0 0 RGB S\n");
 for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".02 W 0 0 .9 RGB S\n");
 for(m=1;m<10;m++)  conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".08 W .9 0 0 RGB S\n");
 for(m=1;m<10;m++)  conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 .9 RGB S\n");
                    conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".08 W .6 0 .6 RGB S\n");
 for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".08 W 0 0 0 RGB S\n");
// y= 0; for(m=0;m<260;m+=6) {x=-2.-.1*m; M(x,y) L(x-.1,y)}
// fprintf(o,".07 W 1 .5 0 RGB S\n");
// y= 0; for(m=3;m<260;m+=6) {x=-2-.1*m; M(x,y) L(x-.1,y)}
// fprintf(o,".07 W 0 .5 1 RGB S\n");
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf tetsheldonmap.eps");   
        system(    "open tetsheldonmap.pdf");
        getchar(); system("killall Preview");
 }

Latex generator of labels

\documentclass[12pt]{article}
\paperwidth 618pt
\paperheight 214pt
\textwidth 1060pt
\textheight 500pt
\topmargin -108pt
\oddsidemargin -70pt
\pagestyle{empty}
\usepackage[usenames]{color}
\usepackage[utf8x]{inputenc}
\usepackage{hyperref} 
\usepackage{graphicx}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \ing {\includegraphics}
\newcommand \rme {\mathrm{e}}
\newcommand \rot {\begin{rotate}} 
\newcommand \ero {\end{rotate}}
\begin{document}
\parindent 0pt
\begin{picture}(202,211)
\put(10,10){\ing{tetsheldonmap}}
\put(2,206){$y$}
\put(2,188){$8$}
\put(2,168){$6$}
\put(2,148){$4$}
\put(2,128){$2$}
\put(2,108){$0$}
\put(-6,88){$-2$}
\put(-6,68){$-4$}
\put(-6,48){$-6$}
\put(-6,28){$-8$}
%\put(0,8){-10}\put(261,0){$-4$}
\put(-1,0){$-30$}
\put( 49,0){$-25$}
\put( 99,0){$-20$}
\put(149,0){$-15$}
\put(199,0){$-10$}
%\put(222,0){$-8$}
%\put(242,0){$-6$}
\put(252,0){$-5$}
%\put(262,0){$-4$}
%\put(282,0){$-2$}
\put(309,0){$0$}
\put(329,0){$2$}
\put(349,0){$4$}
\put(369,0){$6$}
\put(389,0){$8$}
\put(407,0){$10$}
\put(457,0){$15$}
\put(507,0){$20$}
\put(557,0){$25$}
\put(607,1){$x$}
\multiput(24,180)(118,-1){5}{$v\!=\!1.2$}
\multiput(70,180)(118,-1){5}{$u\!=\!2$}
\multiput(112,158)(118,-1){5}{\rot{-50}$v\!=\!1$\ero}
\multiput(102,142)(118,-1){5}{\rot{-35}$v\!=\!0.8$\ero}
\put(336,107){\rot{-11}$v\!=\!0$\ero}
\multiput(206, 92)(119,-22){4}{$v\!=\!-1$}
\multiput(70, 84)(119,-22){4}{$u\!=\!1.2$}
\multiput(110, 76)(119,-22){4}{$v\!=\!-1.4$}
\put(470,200){$u+\mathrm i v \approx 2.0565+1.1445 \,\mathrm i$}
\put(30,20){$u+\mathrm i v \approx 2.2284-1.3508 \,\mathrm i$}
\end{picture}
\end{document}

References

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