Difference between revisions of "File:Olga6map.jpg"

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{{oq|Olga6map.jpg|}}
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Importing image file
  
  +
Complex map of the Olga function,
  +
\(u+\mathrm i v= \mathrm{Olga}(x\!+\!\mathrm i y)\)
  +
  +
Olga is [[scaling function]] for the [[Doya function]]; it is solution of the
  +
[[scaling equation]]
  +
  +
\[\mathrm{Doya}(\mathrm{Olga}(z))=\mathrm e\, \mathrm{Olga}(z)
  +
\]
  +
  +
Olga can be expressed through the [[Tania function]]:
  +
  +
\[
  +
\mathrm{Olga}(z)=\mathrm{Tania}(\ln(z))
  +
\]
  +
  +
This representation is used for the C++ implementation below.
  +
  +
==[[Superfunctions]]==
  +
  +
This image appears as figure 6.4
  +
at page 68 of book «[[Superfunctions]]»<ref>
  +
https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics - 2020/7/28
  +
</ref><ref>
  +
https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). [[Superfunctions]]: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.
  +
</ref><br>
  +
in order to show better the sense of the [[Schroederfunction]]s as analogies of
  +
[[Abelfunction]]s and
  +
[[Scalefunction]]s as analogies of
  +
[[Superfunction]]s.
  +
  +
==[[C++]] generator of map==
  +
//Files [[ado.cin]] and [[conto.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++)
  +
//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 "conto.cin"
  +
z_type ArcTania(z_type z) {return z + log(z) - 1. ;}
  +
z_type ArcTaniap(z_type z) {return 1. + 1./z ;}
  +
z_type TaniaTay(z_type z) { int n; z_type s;
  +
s=1.+z*(.5+z*(1./16.+z*(-1./192.+z*(-1./3072.+z*(1.3/6144.+z*(-4.7/147456.
  +
//+z*(7.3/4128768.) //some reserve term
  +
)))))); DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
  +
z_type TaniaNega(z_type z){int n;z_type s=exp(z-exp(z)+1.);
  +
DO(n,4) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
  +
z_type TaniaBig(z_type z){int n;z_type s=z; s=z-log(s)+1.;
  +
DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
  +
z_type TaniaS(z_type z){int n; z_type s,t=z+z_type(2.,-M_PI);t*=2./9.; t=I*sqrt(t);
  +
s=-1.+t*(3.+t*(-3.+t*(.75+t*(.3+t*(.9/16.+t*(-.3/7.+t*(-12.51/224. //+t*(-.9/28.)
  +
)))))));
  +
DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
  +
z_type Tania(z_type z){ z_type t;
  +
if( fabs(Im(z))< M_PI && Re(z)<-2.51) return TaniaNega(z);
  +
if( abs(z)>7. || Re(z)>3.8 ) return TaniaBig(z);
  +
if( Im(z) > .7 ) return TaniaS(z);
  +
if( Im(z) < -.7) return conj(TaniaS(conj(z)));
  +
return TaniaTay(z);
  +
}
  +
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  +
int M=401,M1=M+1;
  +
int N=603,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("taniacontour.eps","w");ado(o,1620,1620);
  +
//FILE *o;o=fopen("13.eps","w");ado(o,1620,1620);
  +
FILE *o;o=fopen("olga6ma.eps","w");ado(o,1220,1220);
  +
fprintf(o,"610 610 translate\n 100 100 scale\n");
  +
DO(m,M1) X[m]=-6.+.03*(m);
  +
DO(n,300)Y[n]=-6.+.02*n;
  +
Y[300]=-.012;
  +
Y[301]=-.006;
  +
Y[302]= .006;
  +
Y[303]= .012;
  +
for(n=304;n<N1;n++) Y[n]=-6.+.02*(n-3.);
  +
for(m=-6;m<7;m++){if(m==0){M(m,-6.1)L(m,6.1)} else{M(m,-6)L(m,6)}}
  +
for(n=-6;n<7;n++){ M( -6,n)L(6,n)}
  +
fprintf(o,".008 W 0 0 0 RGB 2 setlinecap 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("%5.2f\n",x);
  +
DO(n,N1){y=Y[n]; z=z_type(x,y); // z*=.1;
  +
c=Tania(log(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;}
  +
}}
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=.6;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,".009 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,".009 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,".009 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,".03 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,".03 W 0 0 .9 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".01 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,".03 W 0 0 0 RGB S\n");
  +
  +
  +
// M(-10.*exp(-2.),0)L(-8,0)
  +
M(-exp(-2.),0)L(-8,0)
  +
fprintf(o,"0 setlinecap .01 W 1 1 1 RGB S\n");
  +
  +
y= 0; for(m=1;m<80;m+=4) {x=-.1353352832366127-.1*m; M(x,y) L(x-.08,y)}
  +
fprintf(o,".05 W 1 .5 0 RGB S\n");
  +
y= 0; for(m=3;m<80;m+=4) {x=-.1353352832366127-.1*m; M(x,y) L(x-.08,y)}
  +
fprintf(o,".05 W 0 .5 1 RGB S\n");
  +
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf olga6ma.eps");
  +
system( "open olga6ma.pdf");
  +
getchar(); system("killall Preview");
  +
}
  +
//</pre
  +
  +
==[[Latex]] generator of labels==
  +
<pre>
  +
\documentclass[12pt]{article}
  +
\paperwidth 1268px
  +
\paperheight 1260px
  +
\textwidth 1794px
  +
\textheight 1700px
  +
\topmargin -98px
  +
\oddsidemargin -70px
  +
\usepackage{graphics}
  +
\usepackage{rotating}
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing {\includegraphics}
  +
\newcommand \rmi {\mathrm{i}}
  +
\begin{document}
  +
\parindent 0pt
  +
\sx{1}{\begin{picture}(1252,1230)
  +
%\put(40,20){\ing{suzexD1map}}
  +
%\put(40,20){\ing{ZexD6map}}
  +
\put(50,20){\ing{olga6ma}}
  +
\put(14,1204){\sx{6}{$y$}}
  +
\put(18,1116){\sx{5}{$5$}}
  +
\put(18,1016){\sx{5}{$4$}}
  +
\put(18, 916){\sx{5}{$3$}}
  +
\put(18, 816){\sx{5}{$2$}}
  +
\put(18, 716){\sx{5}{$1$}}
  +
\put(18, 616){\sx{5}{$0$}}
  +
\put(-22, 516){\sx{5}{$-1$}}
  +
\put(-22, 416){\sx{5}{$-2$}}
  +
\put(-22, 316){\sx{5}{$-3$}}
  +
\put(-22, 216){\sx{5}{$-4$}}
  +
\put(-22, 116){\sx{5}{$-5$}}
  +
\put(-22, 016){\sx{5}{$-6$}}
  +
\put( 18, -20){\sx{5}{$-6$}}
  +
\put(118, -20){\sx{5}{$-5$}}
  +
\put(218, -20){\sx{5}{$-4$}}
  +
\put(318, -20){\sx{5}{$-3$}}
  +
\put(418, -20){\sx{5}{$-2$}}
  +
\put(518, -20){\sx{5}{$-1$}}
  +
\put(649, -20){\sx{5}{$0$}}
  +
\put(750, -20){\sx{5}{$1$}}
  +
\put(850, -20){\sx{5}{$2$}}
  +
\put(950, -20){\sx{5}{$3$}}
  +
\put(1050, -20){\sx{5}{$4$}}
  +
\put(1150, -20){\sx{5}{$5$}}
  +
\put(1234, -20){\sx{6}{$x$}}
  +
\put(170, 616){\sx{6}{\bf cut}}
  +
\put(1060, 614){\sx{6}{$v\!=\!0$}}
  +
\put(710, 1086){\sx{6}{\rot{75}$v\!=\!1$\ero}}
  +
\put(660, 222){\sx{6}{\rot{-77}$v\!=\!-1$\ero}}
  +
\put(86,790){\sx{6}{\rot{-29}$v\!=\!2$\ero}}
  +
\put(100, 438){\sx{6}{\rot{28}$v\!=\!-2$\ero}}
  +
%
  +
\put(78, 1050){\sx{6}{\rot{37}$u\!=\!2$\ero}}
  +
\put(460, 668){\sx{6}{\rot{38}$u\!=\!1$\ero}}
  +
  +
\end{picture}}
  +
\end{document}
  +
</pre>
  +
  +
==References==
  +
{{ref}}
  +
  +
{{fer}}
  +
==Keywords==
  +
«[[Abelfunction]]»,
  +
«[[Ernst Schroeder]]»,
  +
«[[Scalefunction]]»,
  +
«[[Schroederfunction]]»,
  +
«[[Superfunctions]]»,
  +
  +
[[Category:Book]]
  +
[[Category:BookMap]]
  +
[[Category:C++]]
  +
[[Category:Latex]]
  +
[[Category:Olga function]]
  +
[[Category:Tania function]]
  +
[[Category:Schroeder function]]
  +
[[Category:Schroederfunction]]
  +
[[Category:Scaling function]]
  +
[[Category:Scalingfunction]]
  +
[[Category:Superfunction]]

Latest revision as of 15:58, 20 August 2025


Complex map of the Olga function, \(u+\mathrm i v= \mathrm{Olga}(x\!+\!\mathrm i y)\)

Olga is scaling function for the Doya function; it is solution of the scaling equation

\[\mathrm{Doya}(\mathrm{Olga}(z))=\mathrm e\, \mathrm{Olga}(z) \]

Olga can be expressed through the Tania function:

\[ \mathrm{Olga}(z)=\mathrm{Tania}(\ln(z)) \]

This representation is used for the C++ implementation below.

Superfunctions

This image appears as figure 6.4 at page 68 of book «Superfunctions»[1][2]
in order to show better the sense of the Schroederfunctions as analogies of Abelfunctions and Scalefunctions as analogies of Superfunctions.

C++ generator of map

//Files ado.cin and 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 "conto.cin"
z_type ArcTania(z_type z) {return z + log(z) - 1. ;}
z_type ArcTaniap(z_type z) {return 1. + 1./z ;}
z_type TaniaTay(z_type z) { int n; z_type s;
s=1.+z*(.5+z*(1./16.+z*(-1./192.+z*(-1./3072.+z*(1.3/6144.+z*(-4.7/147456.
//+z*(7.3/4128768.) //some reserve term
)))))); DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
z_type TaniaNega(z_type z){int n;z_type s=exp(z-exp(z)+1.); 
DO(n,4) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
z_type TaniaBig(z_type z){int n;z_type s=z; s=z-log(s)+1.; 
DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
z_type TaniaS(z_type z){int n; z_type s,t=z+z_type(2.,-M_PI);t*=2./9.; t=I*sqrt(t);
s=-1.+t*(3.+t*(-3.+t*(.75+t*(.3+t*(.9/16.+t*(-.3/7.+t*(-12.51/224. //+t*(-.9/28.)
)))))));
DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; }
z_type Tania(z_type z){ z_type t;
if( fabs(Im(z))< M_PI && Re(z)<-2.51) return TaniaNega(z);
if( abs(z)>7. || Re(z)>3.8 ) return TaniaBig(z);
if( Im(z) > .7 ) return TaniaS(z);
if( Im(z) < -.7) return conj(TaniaS(conj(z)));
return TaniaTay(z);
}

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
 int M=401,M1=M+1;
 int N=603,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("taniacontour.eps","w");ado(o,1620,1620);
//FILE *o;o=fopen("13.eps","w");ado(o,1620,1620);
FILE *o;o=fopen("olga6ma.eps","w");ado(o,1220,1220);
fprintf(o,"610 610 translate\n 100 100 scale\n");
DO(m,M1) X[m]=-6.+.03*(m);
DO(n,300)Y[n]=-6.+.02*n;
        Y[300]=-.012;
        Y[301]=-.006;
        Y[302]= .006;
        Y[303]= .012;
for(n=304;n<N1;n++) Y[n]=-6.+.02*(n-3.);
for(m=-6;m<7;m++){if(m==0){M(m,-6.1)L(m,6.1)} else{M(m,-6)L(m,6)}}
for(n=-6;n<7;n++){     M(  -6,n)L(6,n)}
fprintf(o,".008 W 0 0 0 RGB 2 setlinecap 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("%5.2f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y); // z*=.1;        
 c=Tania(log(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;}
        }}
fprintf(o,"1 setlinejoin 2 setlinecap\n");  p=.6;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,".009 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,".009 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,".009 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,".03 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,".03 W 0 0 .9 RGB S\n");
                   conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".01 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,".03 W 0 0 0 RGB S\n");


// M(-10.*exp(-2.),0)L(-8,0)
M(-exp(-2.),0)L(-8,0)
fprintf(o,"0 setlinecap .01 W 1 1 1 RGB S\n");

y= 0; for(m=1;m<80;m+=4) {x=-.1353352832366127-.1*m; M(x,y) L(x-.08,y)}
fprintf(o,".05 W 1 .5 0 RGB S\n");
y= 0; for(m=3;m<80;m+=4) {x=-.1353352832366127-.1*m; M(x,y) L(x-.08,y)}
fprintf(o,".05 W 0 .5 1 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
       system("epstopdf olga6ma.eps");    
       system(    "open olga6ma.pdf");
       getchar(); system("killall Preview");
}
//</pre

==[[Latex]] generator of labels==
<pre>
\documentclass[12pt]{article} 
\paperwidth 1268px 
\paperheight 1260px 
\textwidth 1794px 
\textheight 1700px 
\topmargin -98px 
\oddsidemargin -70px 
\usepackage{graphics} 
\usepackage{rotating} 
\newcommand \sx {\scalebox} 
\newcommand \rot {\begin{rotate}} 
\newcommand \ero {\end{rotate}} 
\newcommand \ing {\includegraphics} 
\newcommand \rmi {\mathrm{i}} 
\begin{document}  
\parindent 0pt 
\sx{1}{\begin{picture}(1252,1230) 
%\put(40,20){\ing{suzexD1map}} 
%\put(40,20){\ing{ZexD6map}} 
\put(50,20){\ing{olga6ma}} 
\put(14,1204){\sx{6}{$y$}}
\put(18,1116){\sx{5}{$5$}}
\put(18,1016){\sx{5}{$4$}}
\put(18, 916){\sx{5}{$3$}}
\put(18, 816){\sx{5}{$2$}}
\put(18, 716){\sx{5}{$1$}}
\put(18, 616){\sx{5}{$0$}}
\put(-22, 516){\sx{5}{$-1$}}
\put(-22, 416){\sx{5}{$-2$}}
\put(-22, 316){\sx{5}{$-3$}}
\put(-22, 216){\sx{5}{$-4$}}
\put(-22, 116){\sx{5}{$-5$}}
\put(-22, 016){\sx{5}{$-6$}}
\put( 18, -20){\sx{5}{$-6$}}
\put(118, -20){\sx{5}{$-5$}}
\put(218, -20){\sx{5}{$-4$}}
\put(318, -20){\sx{5}{$-3$}}
\put(418, -20){\sx{5}{$-2$}}
\put(518, -20){\sx{5}{$-1$}}
\put(649, -20){\sx{5}{$0$}}
\put(750, -20){\sx{5}{$1$}}
\put(850, -20){\sx{5}{$2$}}
\put(950, -20){\sx{5}{$3$}}
\put(1050, -20){\sx{5}{$4$}}
\put(1150, -20){\sx{5}{$5$}}
\put(1234, -20){\sx{6}{$x$}}
\put(170, 616){\sx{6}{\bf cut}}
\put(1060, 614){\sx{6}{$v\!=\!0$}}
\put(710, 1086){\sx{6}{\rot{75}$v\!=\!1$\ero}} 
\put(660,  222){\sx{6}{\rot{-77}$v\!=\!-1$\ero}} 
\put(86,790){\sx{6}{\rot{-29}$v\!=\!2$\ero}} 
\put(100, 438){\sx{6}{\rot{28}$v\!=\!-2$\ero}} 
%
\put(78, 1050){\sx{6}{\rot{37}$u\!=\!2$\ero}}
\put(460, 668){\sx{6}{\rot{38}$u\!=\!1$\ero}}

\end{picture}} 
\end{document}

References

  1. ↑ https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics - 2020/7/28
  2. ↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.

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