Difference between revisions of "File:Exp09mapT200.jpg"

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(Compex map of the $0.9$th iteration of exponent, $u+\mathrm i v= \exp^{0.9}(x+\mathrm i y)= \mathrm{tet}(0.9+\mathrm{ate}(x+\mathrm i y)$ For the evaluation, the non-integer iterate of exponential is expressed through tetration tet and [...)
 
 
(6 intermediate revisions by the same user not shown)
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==[[C++]] generator of curves==
  
==[[C++]] generator of curves==
  +
  +
// Files [[ado.fin]], [[conto.cin]], [[fsexp.cin]] and [[fslog.cin]] should be loaded in the working directory 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 complex<double> z_type;
  +
#define Re(x) x.real()
  +
#define Im(x) x.imag()
  +
#define I z_type(0.,1.)
  +
#include "conto.cin"
  +
#include "fsexp.cin"
  +
#include "fslog.cin"
  +
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
  +
int M=401,M1=M+1;
  +
int N=403,N1=N+1;
  +
DB X[M1],Y[N1];
  +
DB *g, *f, *w; // w is working array.
  +
g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
  +
f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
  +
w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
  +
char v[M1*N1]; // v is working array
  +
FILE *o;o=fopen("exp09map.eps","w"); ado(o,802,402);
  +
fprintf(o,"401 1 translate\n 100 100 scale\n");
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n");
  +
DO(m,M1) X[m]=-4+.02*(m-.5);
  +
DO(n,N1) { y=0.+.01*(n-.5); if(y>Im(Zo)) break; Y[n]=y; }
  +
Y[n] =Im(Zo)-.00001;
  +
Y[n+1]=Im(Zo)+.00001;
  +
for(m=n+2;m<N1;m++) Y[m]=.01*(m-2-.5);
  +
for(m=-4;m<5;m++){M(m,0) L(m,4) }
  +
for(n=0;n<5;n++){M( -4,n) L(4,n)}
  +
fprintf(o,".006 W 0 0 0 RGB S\n");
  +
DO(m,M1)DO(n,N1){ g[m*N1+n]=999; f[m*N1+n]=999;}
  +
DO(m,M1){x=X[m]; if(m/10*10==m) printf("x=%6.3f\n",x);
  +
DO(n,N1){y=Y[n]; z=z_type(x,y); //if(abs(z+2.)>.019)
  +
c=FSEXP(.9+FSLOG(z));
  +
p=Re(c); q=Im(c);// if(p>-19 && p<19 && ( x<2. || fabs(q)>1.e-12 && fabs(p)>1.e-12) )
  +
{ g[m*N1+n]=p;f[m*N1+n]=q;}
  +
}}
  +
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=2.;q=1;
  +
conto(o,g,w,v,X,Y,M,N, Re(Zo),-p,p);fprintf(o,".03 W 0 1 0 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, Im(Zo),-p,p);fprintf(o,".03 W 0 1 0 RGB S\n");
  +
for(m=-8;m<8;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".007 W 0 .6 0 RGB S\n");
  +
for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".007 W .9 0 0 RGB S\n");
  +
for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".007 W 0 0 .9 RGB S\n");
  +
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-q,q);fprintf(o,".02 W .8 0 0 RGB S\n");
  +
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-q,q);fprintf(o,".02 W 0 0 .8 RGB S\n");
  +
conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".02 W .5 0 .5 RGB S\n");
  +
for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-q,q);fprintf(o,".02 W 0 0 0 RGB S\n");
  +
fprintf(o,"0 setlinejoin 0 setlinecap\n");
  +
  +
M(Re(Zo),Im(Zo))L(-4,Im(Zo)) fprintf(o,"1 1 1 RGB .022 W S\n");
  +
DO(n,40){M(Re(Zo)-.2*n,Im(Zo))L(Re(Zo)-.2*(n+.4),Im(Zo)) }
  +
fprintf(o,"0 0 0 RGB .032 W S\n");
  +
fprintf(o,"showpage\n");
  +
fprintf(o,"%c%cTrailer\n",'%','%');
  +
fclose(o); free(f); free(g); free(w);
  +
system("epstopdf exp09map.eps");
  +
system( "open exp09map.pdf"); //for macintosh
  +
getchar(); system("killall Preview"); // For macintosh
  +
}
  +
   
 
==[[Latex]] generator of labels==
  
==[[Latex]] generator of labels==
  +
  +
% File [[exp09map.pdf]] should be generated with the [[C++]] code above in order to compile the [[Latex]] document below.
  +
  +
<nomathjax>
  +
<poem>
  +
<nowiki>
  +
\documentclass[12pt]{article}
  +
\usepackage{geometry}
  +
\paperwidth 824pt
  +
\paperheight 426pt
  +
\usepackage{graphics}
  +
\usepackage{rotating}
  +
\newcommand \rot {\begin{rotate} }
  +
\newcommand \ero {\end{rotate} }
  +
\textwidth 810pt
  +
\topmargin -105pt
  +
\oddsidemargin -72pt
  +
\pagestyle{empty}
  +
\parindent 0pt
  +
\newcommand \sx {\scalebox}
  +
\begin{document}
  +
\begin{picture}(820,420)
  +
\put(20,20){\includegraphics{exp09map}}
  +
\put(4,412){\sx{2.2}{$y$}}
  +
\put(4,313){\sx{2.2}{$3$}}
  +
\put(4,213){\sx{2.2}{$2$}}
  +
\put(4,113){\sx{2.2}{$1$}}
  +
\put(4,16){\sx{2.2}{$0$}}
  +
\put(2,0){\sx{2.2}{$-4$}}
  +
\put(100,0){\sx{2.2}{$-3$}}
  +
\put(200,0){\sx{2.2}{$-2$}}
  +
\put(300,0){\sx{2.2}{$-1$}}
  +
\put(420,0){\sx{2.2}{$0$}}
  +
\put(520,0){\sx{2.2}{$1$}}
  +
\put(620,0){\sx{2.2}{$2$}}
  +
\put(720,0){\sx{2.2}{$3$}}
  +
\put(810,0){\sx{2.2}{$x$}}
  +
%\put(140,329){\sx{2.4}{$v\!=\!0$}}
  +
\put(200,300){\sx{2}{\rot{-55} $u\!=\!-0.2$ \ero}}
  +
\put(273,307){\sx{2}{\rot{-50} $u\!=\!-0.4$ \ero}}
  +
\put(323,318){\sx{2}{\rot{-56} $u\!=\!-0.6$ \ero}}
  +
\put(353,330){\sx{2}{\rot{-60} $u\!=\!-0.8$ \ero}}
  +
%
  +
\put(264,188){\sx{2}{\rot{ 64} $v\!=\!0.2$ \ero}}
  +
\put(327,173){\sx{2}{\rot{ 73} $v\!=\!0.4$ \ero}}
  +
\put(369,174){\sx{2}{\rot{ 79} $v\!=\!0.6$ \ero}}
  +
\put(398,177){\sx{2}{\rot{ 79} $v\!=\!0.8$ \ero}}
  +
\put(422,180){\sx{2.2}{\rot{ 82} $v\!=\!1$ \ero}}
  +
\put(450,178){\sx{1.9}{\rot{ 79} $v\!=\!\Im(L)$ \ero}}
  +
\put(495,178){\sx{2}{\rot{ 90} $v\!=\!2$ \ero}}
  +
\put(539,178){\sx{2}{\rot{ 90} $v\!=\!3$ \ero}}
  +
\put(570,178){\sx{2}{\rot{ 90} $v\!=\!4$ \ero}}
  +
%\put(40,180){\sx{2}{$u=0$}}
  +
\put(62,226){\sx{2.4}{\rot{22}$v\!=\!0$\ero}}
  +
%
  +
\put(204,38){\sx{2.4}{\rot{ 66} $u\!=\!0$ \ero}}
  +
\put(303,22){\sx{2}{\rot{ 70} $u\!=\!0.2$ \ero}}
  +
\put(332,22){\sx{2}{\rot{ 69} $u\!=\!\Re(L)$ \ero}}
  +
\put(356,22){\sx{2}{\rot{ 70} $u\!=\!0.4$ \ero}}
  +
\put(386,22){\sx{2}{\rot{ 70} $u\!=\!0.6$ \ero}}
  +
\put(412,22){\sx{2}{\rot{ 70} $u\!=\!0.8$ \ero}}
  +
\put(435,22){\sx{2.2}{\rot{ 70} $u\!=\!1$ \ero}}
  +
\put(506,22){\sx{2.2}{\rot{ 73} $u\!=\!2$ \ero}}
  +
\put(546,22){\sx{2.2}{\rot{ 73} $u\!=\!3$ \ero}}
  +
\put(580,22){\sx{2.2}{\rot{ 73} $u\!=\!4$ \ero}}
  +
\put(56,150){\sx{2.6}{\bf cut}}
  +
\end{picture}
  +
\end{document}
  +
</nowiki>
  +
</poem>
  +
</nomathjax>
  +
==References==
  +
<references/>
  +
http://www.ams.org/journals/bull/1993-29-02/S0273-0979-1993-00432-4/S0273-0979-1993-00432-4.pdf Walter Bergweiler. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29 (1993), 151-188.
  +
  +
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html<br>
  +
http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf<br>
  +
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D. Kouznetsov. Solution of F(x+1)=exp(F(x)) in complex z-plane. 78, (2009), 1647-1670
  +
  +
http://www.ils.uec.ac.jp.jp/~dima/PAPERS/2009vladie.pdf (English) <br>
  +
http://mizugadro.mydns.jp/PAPERS/2009vladir.pdf (Russian version) <br>
  +
http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf (English)
  +
D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.
  +
  +
http://reference.wolfram.com/mathematica/ref/Nest.html Nest, Wolfram Mathematica 9 Documentation center, 2013
   
 
[[Category:Arctetration]]
  
[[Category:Arctetration]]
 
[[Category:Complex map]]
  
[[Category:Complex map]]
−
]]Category:C++]]
 +
[[Category:C++]]
 
[[Category:Exponent]]
  
[[Category:Exponent]]
 
[[Category:Iteration]]
  
[[Category:Iteration]]
  +
[[Category:Iteration of exp]]
 
[[Category:Latex]]
  
[[Category:Latex]]
 
[[Category:Tetration]]
  
[[Category:Tetration]]

Latest revision as of 12:28, 28 July 2013

Compex map of the $0.9$th iteration of exponent,

$u+\mathrm i v= \exp^{0.9}(x+\mathrm i y)= \mathrm{tet}(0.9+\mathrm{ate}(x+\mathrm i y)$

For the evaluation, the non-integer iterate of exponential is expressed through tetration tet and arctetration ate. The complex double implementations FSEXP and FSLOG are used in the C++ code below.

C++ generator of curves

// Files ado.fin, conto.cin, fsexp.cin and fslog.cin should be loaded in the working directory 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 complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"
#include "fsexp.cin"
#include "fslog.cin"
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
int M=401,M1=M+1;
int N=403,N1=N+1;
DB X[M1],Y[N1];
DB *g, *f, *w; // w is working array.
g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB)));
char v[M1*N1]; // v is working array
FILE *o;o=fopen("exp09map.eps","w"); ado(o,802,402);
fprintf(o,"401 1 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
DO(m,M1) X[m]=-4+.02*(m-.5);
DO(n,N1) { y=0.+.01*(n-.5); if(y>Im(Zo)) break; Y[n]=y; }
Y[n]  =Im(Zo)-.00001;
Y[n+1]=Im(Zo)+.00001;
for(m=n+2;m<N1;m++) Y[m]=.01*(m-2-.5);
for(m=-4;m<5;m++){M(m,0) L(m,4)  }
for(n=0;n<5;n++){M(  -4,n) L(4,n)}
 fprintf(o,".006 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){      g[m*N1+n]=999; f[m*N1+n]=999;}
DO(m,M1){x=X[m]; if(m/10*10==m) printf("x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y); //if(abs(z+2.)>.019)
c=FSEXP(.9+FSLOG(z));
 p=Re(c); q=Im(c);// if(p>-19 && p<19 && ( x<2. ||  fabs(q)>1.e-12 && fabs(p)>1.e-12) )
                  { g[m*N1+n]=p;f[m*N1+n]=q;}
       }}
fprintf(o,"1 setlinejoin 1 setlinecap\n");   p=2.;q=1;
conto(o,g,w,v,X,Y,M,N, Re(Zo),-p,p);fprintf(o,".03 W 0 1 0 RGB S\n");
conto(o,f,w,v,X,Y,M,N, Im(Zo),-p,p);fprintf(o,".03 W 0 1 0 RGB S\n");
for(m=-8;m<8;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".007 W 0 .6 0 RGB S\n");
for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".007 W .9 0 0 RGB S\n");
for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".007 W 0 0 .9 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-q,q);fprintf(o,".02 W .8 0 0 RGB S\n");
for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-q,q);fprintf(o,".02 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0.  ),-p,p); fprintf(o,".02 W .5 0 .5 RGB S\n");
for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-q,q);fprintf(o,".02 W 0 0 0 RGB S\n");
fprintf(o,"0 setlinejoin 0 setlinecap\n");

M(Re(Zo),Im(Zo))L(-4,Im(Zo)) fprintf(o,"1 1 1 RGB .022 W S\n");
DO(n,40){M(Re(Zo)-.2*n,Im(Zo))L(Re(Zo)-.2*(n+.4),Im(Zo)) }
 fprintf(o,"0 0 0 RGB .032 W S\n");
fprintf(o,"showpage\n");
fprintf(o,"%c%cTrailer\n",'%','%');
fclose(o);  free(f); free(g); free(w);
      system("epstopdf exp09map.eps"); 
      system(    "open exp09map.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}


Latex generator of labels

% File exp09map.pdf should be generated with the C++ code above in order to compile the Latex document below.


 \documentclass[12pt]{article}
 \usepackage{geometry}
 \paperwidth 824pt
 \paperheight 426pt
 \usepackage{graphics}
 \usepackage{rotating}
 \newcommand \rot {\begin{rotate} }
 \newcommand \ero {\end{rotate} }
 \textwidth 810pt
 \topmargin -105pt
 \oddsidemargin -72pt
 \pagestyle{empty}
 \parindent 0pt
 \newcommand \sx {\scalebox}
\begin{document}
\begin{picture}(820,420)
\put(20,20){\includegraphics{exp09map}}
\put(4,412){\sx{2.2}{$y$}}
\put(4,313){\sx{2.2}{$3$}}
\put(4,213){\sx{2.2}{$2$}}
\put(4,113){\sx{2.2}{$1$}}
\put(4,16){\sx{2.2}{$0$}}
\put(2,0){\sx{2.2}{$-4$}}
\put(100,0){\sx{2.2}{$-3$}}
\put(200,0){\sx{2.2}{$-2$}}
\put(300,0){\sx{2.2}{$-1$}}
\put(420,0){\sx{2.2}{$0$}}
\put(520,0){\sx{2.2}{$1$}}
\put(620,0){\sx{2.2}{$2$}}
\put(720,0){\sx{2.2}{$3$}}
\put(810,0){\sx{2.2}{$x$}}
%\put(140,329){\sx{2.4}{$v\!=\!0$}}
\put(200,300){\sx{2}{\rot{-55} $u\!=\!-0.2$ \ero}}
\put(273,307){\sx{2}{\rot{-50} $u\!=\!-0.4$ \ero}}
\put(323,318){\sx{2}{\rot{-56} $u\!=\!-0.6$ \ero}}
\put(353,330){\sx{2}{\rot{-60} $u\!=\!-0.8$ \ero}}
%
\put(264,188){\sx{2}{\rot{ 64} $v\!=\!0.2$ \ero}}
\put(327,173){\sx{2}{\rot{ 73} $v\!=\!0.4$ \ero}}
\put(369,174){\sx{2}{\rot{ 79} $v\!=\!0.6$ \ero}}
\put(398,177){\sx{2}{\rot{ 79} $v\!=\!0.8$ \ero}}
\put(422,180){\sx{2.2}{\rot{ 82} $v\!=\!1$ \ero}}
\put(450,178){\sx{1.9}{\rot{ 79} $v\!=\!\Im(L)$ \ero}}
\put(495,178){\sx{2}{\rot{ 90} $v\!=\!2$ \ero}}
\put(539,178){\sx{2}{\rot{ 90} $v\!=\!3$ \ero}}
\put(570,178){\sx{2}{\rot{ 90} $v\!=\!4$ \ero}}
%\put(40,180){\sx{2}{$u=0$}}
\put(62,226){\sx{2.4}{\rot{22}$v\!=\!0$\ero}}
%
\put(204,38){\sx{2.4}{\rot{ 66} $u\!=\!0$ \ero}}
\put(303,22){\sx{2}{\rot{ 70} $u\!=\!0.2$ \ero}}
\put(332,22){\sx{2}{\rot{ 69} $u\!=\!\Re(L)$ \ero}}
\put(356,22){\sx{2}{\rot{ 70} $u\!=\!0.4$ \ero}}
\put(386,22){\sx{2}{\rot{ 70} $u\!=\!0.6$ \ero}}
\put(412,22){\sx{2}{\rot{ 70} $u\!=\!0.8$ \ero}}
\put(435,22){\sx{2.2}{\rot{ 70} $u\!=\!1$ \ero}}
\put(506,22){\sx{2.2}{\rot{ 73} $u\!=\!2$ \ero}}
\put(546,22){\sx{2.2}{\rot{ 73} $u\!=\!3$ \ero}}
\put(580,22){\sx{2.2}{\rot{ 73} $u\!=\!4$ \ero}}
\put(56,150){\sx{2.6}{\bf cut}}
\end{picture}
\end{document}

References

http://www.ams.org/journals/bull/1993-29-02/S0273-0979-1993-00432-4/S0273-0979-1993-00432-4.pdf Walter Bergweiler. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29 (1993), 151-188.

http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf
http://mizugadro.mydns.jp/PAPERS/2009analuxpRepri.pdf D. Kouznetsov. Solution of F(x+1)=exp(F(x)) in complex z-plane. 78, (2009), 1647-1670

http://www.ils.uec.ac.jp.jp/~dima/PAPERS/2009vladie.pdf (English)
http://mizugadro.mydns.jp/PAPERS/2009vladir.pdf (Russian version)
http://mizugadro.mydns.jp/PAPERS/2010vladie.pdf (English) D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.

http://reference.wolfram.com/mathematica/ref/Nest.html Nest, Wolfram Mathematica 9 Documentation center, 2013

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current01:18, 27 July 2013Thumbnail for version as of 01:18, 27 July 20132,281 × 1,179 (1.23 MB)T (talk | contribs)Compex map of the $0.9$th iteration of exponent, $u+\mathrm i v= \exp^{0.9}(x+\mathrm i y)= \mathrm{tet}(0.9+\mathrm{ate}(x+\mathrm i y)$ For the evaluation, the non-integer iterate of exponential is expressed through tetration tet and [...

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