Difference between revisions of "File:Exm01mapT200.jpg"

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For the evaluation, the non-integer iterate of exponential is expressed through [[tetration]] tet and [[arctetration]] ate.
  
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.
+
The complex double implementations [[FSEXP]] and [[FSLOG]] of these functions are used in the [[C++]] code below.
   
 
==[[C++]] generator of curves==
  
==[[C++]] generator of curves==
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\textwidth 810pt
  
\textwidth 810pt
 
\topmargin -105pt
  
\topmargin -105pt
−
\oddsidemargin -72pt
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\oddsidemargin -74pt
 
\pagestyle{empty}
  
\pagestyle{empty}
 
\parindent 0pt
  
\parindent 0pt
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\put(620,0){\sx{2.2}{$2$}}
  
\put(620,0){\sx{2.2}{$2$}}
 
\put(720,0){\sx{2.2}{$3$}}
  
\put(720,0){\sx{2.2}{$3$}}
−
\put(810,0){\sx{2.2}{$x$}}
 +
\put(811,0){\sx{2.2}{$x$}}
 
%\put(140,329){\sx{2.4}{$v\!=\!0$}}
  
%\put(140,329){\sx{2.4}{$v\!=\!0$}}
 
\put(164,368){\sx{2.6}{\rot{18} $v\!=\!4$ \ero}}
  
\put(164,368){\sx{2.6}{\rot{18} $v\!=\!4$ \ero}}
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%</nowiki></nomathjax></poem>
  
%</nowiki></nomathjax></poem>
   
  +
==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/2010vladie.pdf (English) <br>
  +
http://mizugadro.mydns.jp/PAPERS/2009vladir.pdf (Russian version) <br>
  +
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]]
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[[Category:Exponent]]
  
[[Category:Exponent]]
 
[[Category:Iteration]]
  
[[Category:Iteration]]
  +
[[Category:Iteration of exp]]
 
[[Category:Latex]]
  
[[Category:Latex]]
 
[[Category:Logarithm]]
  
[[Category:Logarithm]]

Latest revision as of 12:52, 28 July 2013

Complex map of the $-0.1$th iteration of exponent,

$u+\mathrm i v= \exp^{-0.1}(x+\mathrm i y)=\ln^{0.1}(x+\mathrm i y)= \mathrm{tet}(-0.1+\mathrm{ate}(x+\mathrm i y))$

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

C++ generator of curves

// Files ado.cin, 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("exm01map.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(-.1+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");

DO(n,16) { M(-2.3-.2*n , 0) L(-2.3-.2*(n+.4),0) }
 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 exm01map.eps"); 
      system(    "open exm01map.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}


Latex generator of labels

%


\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 -74pt
\pagestyle{empty}
\parindent 0pt
\newcommand \sx {\scalebox}
\begin{document}
\begin{picture}(820,420)
\put(20,20){\includegraphics{exm01map}}
\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(811,0){\sx{2.2}{$x$}}
%\put(140,329){\sx{2.4}{$v\!=\!0$}}
\put(164,368){\sx{2.6}{\rot{18} $v\!=\!4$ \ero}}
\put(546,349){\sx{2.5}{\rot{ 9} $v\!=\!3$ \ero}}
\put(550,233){\sx{2.5}{\rot{ 8} $v\!=\!2$ \ero}}
\put(642,166){\sx{2.5}{\rot{ 5} $v\!=\!\Im(L)$ \ero}}
\put(558,122){\sx{2.5}{\rot{ 5} $v\!=\!1$ \ero}}
\put(554,14){\sx{2.5}{\rot{-.01} $v\!=\!0$ \ero}}
%
\put( 64,53){\sx{2.5}{\rot{-23} $v\!=\!3$ \ero}}
\put(192,70){\sx{2.5}{\rot{73} $v\!=\!2$ \ero}}
\put(253,100){\sx{2.5}{\rot{-78} $u\!=\!-2$ \ero}}
\put(330,100){\sx{2.4}{\rot{-84} $u\!=\!-1$ \ero}}
\put(422,100){\sx{2.5}{\rot{-86} $u\!=\!0$ \ero}}
\put(417,328){\sx{2.4}{\rot{-79} $u\!=\!\Re(L)$ \ero}}
\put(525,100){\sx{2.5}{\rot{-89} $u\!=\!1$ \ero}}
\put(635,100){\sx{2.5}{\rot{-89} $u\!=\!2$ \ero}}
\put(753,100){\sx{2.5}{\rot{-89.6} $u\!=\!3$ \ero}}
%
\put(53,150){\sx{2.6}{\bf cut}}
\put(44,14){\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/2010vladie.pdf (English)
http://mizugadro.mydns.jp/PAPERS/2009vladir.pdf (Russian version)
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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current12:51, 28 July 2013Thumbnail for version as of 12:51, 28 July 20132,281 × 1,179 (834 KB)T (talk | contribs)resolution
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20:49, 27 July 2013Thumbnail for version as of 20:49, 27 July 20132,281 × 1,179 (833 KB)T (talk | contribs)Compex map of the $-0.1$th iteration of exponent, $u+\mathrm i v= \exp^{-0.1}(x+\mathrm i y)=\ln^{0.1}(x+\mathrm i y)= \mathrm{tet}(-0.1+\mathrm{ate}(x+\mathrm i y)$ For the evaluation, the non-integer iterate of exponential is expressed thr...

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