Difference between revisions of "File:Aupower2map.jpg"

Complex map of the Abelfunction \(G\) of the power transfer function (quadratic function) \(T(z)\!=\!z^2\) is shown with

\(u\!+\!\mathrm i v=G(x\!+\!\mathrm i y)\)

\(G(z)=\log_2(\ln(z))=\ln^2(z)/\ln(2)\)

is solution of the Abel equation

\(G(T(z))=G(z)+1\)

Abelfunction \(G\) is inverse of the superpower function

\(F(z)=\exp(2^z)\)

which is solution of the transfer equation

\(F(z\!+\!1)=T(F(z))\)

for the quadratic transfer function \(T\) at base \(2\).

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++)
#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"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=502,M1=M+1;
int N=502,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("aupower2ma.eps","w");ado(o,1020,1020);
fprintf(o,"510 510 translate\n 100 100 scale\n");
DO(m,M1) X[m]=-5.+.02*(m-.5);
for(n=0;n<250;n++) Y[n]=-5.+.02*(n);
Y[250]=-.006;
Y[251]= .006;
for(n=252;n<N1;n++)Y[n]=-5.+.02*(n-1);
//DO(m,M1) X[m]=Y[m];

for(m=-5;m<6;m++){if(m==0){M(m,-5.1)L(m,5.1)} else{M(m,-5)L(m,5)}}
for(n=-5;n<6;n++){     M(  -5,n)L(5,n)}
fprintf(o,".01 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){g[m*N1+n]=999999; f[m*N1+n]=999999;}
DO(m,M1){x=X[m]; //printf("%5.2f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
//       c=exp(pow(2.,z));
       c=log(log(z))/log(2.);
      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 1 setlinecap\n"); p=1;q=.5;
for(m=-11;m<11;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".013 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,".012 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,".012 W 0 0 .9 RGB S\n");
for(m=1;m<11;m++)  conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".028 W .9 0 0 RGB S\n");
for(m=1;m<11;m++)  conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".028 W 0 0 .9 RGB S\n");
                conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".028 W .6 0 .6 RGB S\n");
for(m=-10;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".024 W 0 0 0 RGB S\n");
             m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".024 W 0 0 0 RGB S\n");
for(m=1;m<11;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".024 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
      system("epstopdf aupower2ma.eps");
      system(    "open aupower2ma.pdf");            
      getchar(); system("killall Preview");//for mac
}

Latex generator of labels

\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{geometry}
\usepackage{rotating} 
\paperwidth 1044px
\paperheight 1036px
\topmargin -98px
\oddsidemargin -90px
\textwidth 2000px
\textheight 2000px
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\begin{document}
\begin{picture}(1030,1026) \normalsize
\put(30,20){\ing{"aupower2ma"}}
\put(16,1016){\sx{3}{$y$}}
\put(16,918){\sx{3}{$4$}}
\put(16,818){\sx{3}{$3$}}
\put(16,718){\sx{3}{$2$}}
\put(16,618){\sx{3}{$1$}}
\put(16,518){\sx{3}{$0$}}
\put(-10,418){\sx{3}{$-1$}}
\put(-10,318){\sx{3}{$-2$}}
\put(-10,218){\sx{3}{$-3$}}
\put(-10,118){\sx{3}{$-4$}}
\put(-10,18){\sx{3}{$-5$}}
\put(10,-1){\sx{3}{$-5$}}
\put(110,-1){\sx{3}{$-4$}}
\put(210,-1){\sx{3}{$-3$}}
\put(310,-1){\sx{3}{$-2$}}
\put(410,-1){\sx{3}{$-1$}}
\put(534,-1){\sx{3}{$0$}}
\put(634,-1){\sx{3}{$1$}}
\put(734,-1){\sx{3}{$2$}}
\put(834,-1){\sx{3}{$3$}}
\put(934,-1){\sx{3}{$4$}}
\put(1028,-1){\sx{3.1}{$x$}}
\put(150,522){\rot{0}{ \sx{3}{\bf cut}} \ero} 
%\put(150,68){\rot{0.}{ \sx{3}{$v\!=\!0$}} \ero}
%\put(790,522){\rot{0}{ \sx{3}{$v\!=\!0$}} \ero}
\put(246,422){\rot{34}{ \sx{3}{$u\!=\!1.6$}} \ero}
\put(332,344){\rot{54}{ \sx{3}{$u\!=\!1.4$}} \ero}%%
\put(295,900){\rot{84}{ \sx{3}{$u\!=\!1.4$}} \ero}%
\put(430,882){\rot{63}{ \sx{3}{$u\!=\!1.2$}} \ero}
\put(526,845){\rot{50}{ \sx{3}{$u\!=\!1$}} \ero}
\put(490,662){\rot{-5}{ \sx{3}{$v\!=\!2$}} \ero}%%
\put(734,742){\rot{90}{ \sx{3}{$v\!=\!1$}} \ero}
\put(790,740){\rot{75}{ \sx{3}{$v\!=\!0.8$}} \ero}
\put(846,710){\rot{56}{ \sx{3}{$v\!=\!0.6$}} \ero}
\put(900,664){\rot{39}{ \sx{3}{$v\!=\!0.4$}} \ero}
\put(930,600){\rot{17}{ \sx{3}{$v\!=\!0.2$}} \ero}
\put(950,523){\rot{0}{ \sx{3}{$v\!=\!0$}} \ero}
\put(910,453){\rot{-20}{ \sx{3}{$v\!=\!-0.2$}} \ero}
\put(733,244){\rot{90}{ \sx{3}{$v\!=\!-1$}} \ero}
\put(490,388){\rot{6}{ \sx{3}{$v\!=\!-2$}} \ero}
%
\put(746,432){\rot{65}{ \sx{3}{$u\!=\!-0.2$}} \ero}
\put(778,420){\rot{57}{ \sx{3}{$u\!=\!0$}} \ero}
\put(800,390){\rot{51}{ \sx{3}{$u\!=\!0.2$}} \ero}
\put(846,358){\rot{50}{ \sx{3}{$u\!=\!0.4$}} \ero}
\put(902,308){\rot{47}{ \sx{3}{$u\!=\!0.6$}} \ero}
\put(965,227){\rot{41}{ \sx{3}{$u\!=\!0.8$}} \ero}
\end{picture}
\end{document}

References

Keywords

«[[]]», «Abelpower», «Power function», «Superfunctions», «Superpower», «Table of superfunctions»,