Difference between revisions of "File:B271t.png"

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[[Complex map]] of tetration to base $\mathrm e$, isolines of real and imaginary parts of
Importing image file
 
  +
: $f\!=\!\mathrm{tet}(x + \mathrm i y)$ are shown in the $x$,$y$ plane,
  +
range:
  +
$~-4\! \le\! x \le 4~$;
  +
$~-4\! \le\! y \le 4~$.
  +
  +
Levels
  +
$u\!=\!\Re(f)\!=\!\mathrm {const}$ and
  +
$v\!=\!\Im(f)\!=\!\mathrm {const}$ are plotted; integer values correspond to the thick lines.
  +
  +
==C++ generator of curves==
  +
// Files [[conto.cin]], [[ado.cin]], [[fsexp.cin]], [[plofu.cin]] should be loaded to the working directory for the compilation of 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=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
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FILE *o;o=fopen("b271t0.eps","w"); ado(o,87,87);
  +
fprintf(o,"46 45 translate\n 10 10 scale\n");
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DO(m,M1) X[m]=-4.+.02*(m-.5);
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DO(n,N1) Y[n]=-4.+.02*(n-.5);
  +
for(m=-4;m<5;m++) {M(m,-4)L(m,4)}
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for(n=-4;n<5;n++) {M( -4,n)L(4,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
  +
fprintf(o,"/adobe-Roman findfont .6 scalefont setfont\n");
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DO(m,M1)DO(n,N1){ g[m*N1+n]=9999;
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f[m*N1+n]=9999;}
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DB b=sqrt(2);
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DO(m,M1){x=X[m]; printf("x=%6.3f\n",x);
  +
DO(n,N1){y=Y[n]; z=z_type(x,y); if(abs(z+2.)>.04)
  +
{ c=FSEXP(z);
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// c=FSLOG(z);
  +
p=Re(c); q=Im(c);
  +
if(p>-9999 && p<9999 && fabs(q)>1.e-12) g[m*N1+n]=p;
  +
if(q>-9999 && q<9999 && fabs(q)>1.e-12) f[m*N1+n]=q;
  +
}
  +
}}
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n");
  +
p=2.; q=1.1;;
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#include "plofu.cin"
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fprintf(o,"0 setlinejoin 0 setlinecap\n");
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fprintf(o,"showpage\n");
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fprintf(o,"%cTrailer\n",'%');
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fclose(o);
  +
system("epstopdf b271t0.eps");
  +
system( "open b271t0.pdf"); //for macintosh
  +
getchar(); system("killall Preview"); // For macintosh
  +
}
  +
  +
==Latex generator of labels==
  +
<nowiki>
  +
\documentclass[12pt]{article} %<br>
  +
\paperheight 680px %<br>
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\paperwidth 772px %<br>
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\textwidth 1294px %<br>
  +
\textheight 1200px %<br>
  +
\topmargin -110px %<br>
  +
\oddsidemargin -80px %<br>
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\usepackage{graphics} %<br>
  +
\usepackage{rotating} %<br>
  +
\newcommand \sx {\scalebox} %<br>
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\newcommand \rot {\begin{rotate}} %<br>
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\newcommand \ero {\end{rotate}} %<br>
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\newcommand \ing {\includegraphics} %<br>
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\newcommand \rmi {\mathrm{i}} %<br>
  +
\begin{document} %<br>
  +
\newcommand \zoomax { %<br>
  +
\put(2, 82.6){\sx{.6}{$y$}} %<br>
  +
\put(2, 63.3){\sx{.5}{$2$}} %<br>
  +
\put(2, 43.3){\sx{.5}{$0$}} %<br>
  +
\put(-1, 23.3){\sx{.5}{$-\!2$}} %<br>
  +
\put(22, 0.8){\sx{.5}{$-\!2$}} %<br>
  +
\put(45, 0.8){\sx{.5}{$0$}} %<br>
  +
\put(65, 0.8){\sx{.5}{$2$}} %<br>
  +
\put(84,1){\sx{.5}{$x$}} %<br>
  +
} %<br>
  +
\parindent 0pt %<br>
  +
\sx{8}{\begin{picture}(86,86) \put(0,0){\ing{b271t0}} %<br>
  +
\zoomax %<br>
  +
\put(7,44){\sx{.4}{\bf cut}} %<br>
  +
\put(7,81){\sx{.4}{$f\approx L\approx0.318+1.337i$}} %<br>
  +
\put(6,6){\sx{.4}{$f\approx L^*\approx0.318-1.337i$}} %<br>
  +
\put(87,78){\sx{.31}{$u\!=\!0.4$}} %<br>
  +
\put(87,74){\sx{.31}{$v\!=\!1.2$}} %<br>
  +
\put(87,68){\sx{.32}{$v\!=\!1$}} %<br>
  +
\put(87,56){\sx{.32}{$v\!=\!0$}} %<br>
  +
\put(87,44){\sx{.32}{$v\!=\!0$}} %<br>
  +
\put(87,33){\sx{.31}{$v\!=\!0$}} %<br>
  +
\put(87,21){\sx{.31}{$v\!=\!-\!1$}} %<br>
  +
\put(86,14){\sx{.29}{$v\!=\!-\!1.2$}} %<br>
  +
\put(87, 11){\sx{.30}{$u\!=\!0.4$}} %<br>
  +
\put(10.2,66){\sx{.3}{\rot{2}$v\!=\!1.4$\ero}} \put(55.2,77){\sx{.3}{\rot{2}$v\!=\!1.4$\ero}} %<br>
  +
\put(10,22){\sx{.3}{\rot{-2}$v\!=\!-\!1.4$\ero}} \put(55,12){\sx{.3}{\rot{-2}$v\!=\!-\!1.4$\ero}} %<br>
  +
\put(22,64){\sx{.3}{\rot{15}$u\!=\!0.2$\ero}} \put(41,71){\sx{.3}{\rot{16}$u\!=\!0.4$\ero}} %<br>
  +
\put(22,25){\sx{.3}{\rot{-15}$u\!=\!0.2$\ero}} \put(41,18){\sx{.3}{\rot{-16}$u\!=\!0.4$\ero}} %<br>
  +
\end{picture}} %<br>
  +
\end{document}
  +
</nowiki>
  +
==References==
  +
  +
http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf
  +
D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670. <!--
  +
http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
  +
!-->
  +
  +
http://tori.ils.uec.ac.jp/PAPERS/2010vladie.pdf
  +
D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.
  +
  +
Copyleft 2009–2011 by Dmitrii Kouznetsov:
  +
Use for free, attribute the source. The generators are available and will be uploaded upon request.
  +
  +
[[Category:Complex maps]]
  +
[[Category:Tetration]]

Revision as of 09:39, 21 June 2013

Complex map of tetration to base $\mathrm e$, isolines of real and imaginary parts of

$f\!=\!\mathrm{tet}(x + \mathrm i y)$ are shown in the $x$,$y$ plane,

range: $~-4\! \le\! x \le 4~$; $~-4\! \le\! y \le 4~$.

Levels $u\!=\!\Re(f)\!=\!\mathrm {const}$ and $v\!=\!\Im(f)\!=\!\mathrm {const}$ are plotted; integer values correspond to the thick lines.

C++ generator of curves

// Files conto.cin, ado.cin, fsexp.cin, plofu.cin should be loaded to the working directory for the compilation of 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=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("b271t0.eps","w");  ado(o,87,87);
fprintf(o,"46 45 translate\n 10 10 scale\n");
DO(m,M1) X[m]=-4.+.02*(m-.5);
DO(n,N1) Y[n]=-4.+.02*(n-.5);
for(m=-4;m<5;m++) {M(m,-4)L(m,4)}
for(n=-4;n<5;n++) {M(  -4,n)L(4,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
fprintf(o,"/adobe-Roman findfont .6 scalefont setfont\n");
DO(m,M1)DO(n,N1){      g[m*N1+n]=9999;
                       f[m*N1+n]=9999;}
DB b=sqrt(2);
DO(m,M1){x=X[m]; printf("x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y); if(abs(z+2.)>.04)
       { c=FSEXP(z);
//       c=FSLOG(z);
         p=Re(c); q=Im(c);
         if(p>-9999 && p<9999 && fabs(q)>1.e-12) g[m*N1+n]=p;
         if(q>-9999 && q<9999 && fabs(q)>1.e-12) f[m*N1+n]=q;
       }
        }}
fprintf(o,"1 setlinejoin 2 setlinecap\n");
p=2.; q=1.1;;
#include "plofu.cin"
fprintf(o,"0 setlinejoin 0 setlinecap\n");
fprintf(o,"showpage\n");
fprintf(o,"%cTrailer\n",'%');
fclose(o);
       system("epstopdf b271t0.eps"); 
       system(    "open b271t0.pdf"); //for macintosh
       getchar(); system("killall Preview"); // For macintosh

}

Latex generator of labels

\documentclass[12pt]{article} %<br> \paperheight 680px %<br> \paperwidth 772px %<br> \textwidth 1294px %<br> \textheight 1200px %<br> \topmargin -110px %<br> \oddsidemargin -80px %<br> \usepackage{graphics} %<br> \usepackage{rotating} %<br> \newcommand \sx {\scalebox} %<br> \newcommand \rot {\begin{rotate}} %<br> \newcommand \ero {\end{rotate}} %<br> \newcommand \ing {\includegraphics} %<br> \newcommand \rmi {\mathrm{i}} %<br> \begin{document} %<br> \newcommand \zoomax { %<br> \put(2, 82.6){\sx{.6}{$y$}} %<br> \put(2, 63.3){\sx{.5}{$2$}} %<br> \put(2, 43.3){\sx{.5}{$0$}} %<br> \put(-1, 23.3){\sx{.5}{$-\!2$}} %<br> \put(22, 0.8){\sx{.5}{$-\!2$}} %<br> \put(45, 0.8){\sx{.5}{$0$}} %<br> \put(65, 0.8){\sx{.5}{$2$}} %<br> \put(84,1){\sx{.5}{$x$}} %<br> } %<br> \parindent 0pt %<br> \sx{8}{\begin{picture}(86,86) \put(0,0){\ing{b271t0}} %<br> \zoomax %<br> \put(7,44){\sx{.4}{\bf cut}} %<br> \put(7,81){\sx{.4}{$f\approx L\approx0.318+1.337i$}} %<br> \put(6,6){\sx{.4}{$f\approx L^*\approx0.318-1.337i$}} %<br> \put(87,78){\sx{.31}{$u\!=\!0.4$}} %<br> \put(87,74){\sx{.31}{$v\!=\!1.2$}} %<br> \put(87,68){\sx{.32}{$v\!=\!1$}} %<br> \put(87,56){\sx{.32}{$v\!=\!0$}} %<br> \put(87,44){\sx{.32}{$v\!=\!0$}} %<br> \put(87,33){\sx{.31}{$v\!=\!0$}} %<br> \put(87,21){\sx{.31}{$v\!=\!-\!1$}} %<br> \put(86,14){\sx{.29}{$v\!=\!-\!1.2$}} %<br> \put(87, 11){\sx{.30}{$u\!=\!0.4$}} %<br> \put(10.2,66){\sx{.3}{\rot{2}$v\!=\!1.4$\ero}} \put(55.2,77){\sx{.3}{\rot{2}$v\!=\!1.4$\ero}} %<br> \put(10,22){\sx{.3}{\rot{-2}$v\!=\!-\!1.4$\ero}} \put(55,12){\sx{.3}{\rot{-2}$v\!=\!-\!1.4$\ero}} %<br> \put(22,64){\sx{.3}{\rot{15}$u\!=\!0.2$\ero}} \put(41,71){\sx{.3}{\rot{16}$u\!=\!0.4$\ero}} %<br> \put(22,25){\sx{.3}{\rot{-15}$u\!=\!0.2$\ero}} \put(41,18){\sx{.3}{\rot{-16}$u\!=\!0.4$\ero}} %<br> \end{picture}} %<br> \end{document}

References

http://www.ils.uec.ac.jp/~dima/PAPERS/2009analuxpRepri.pdf D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathematics of Computation, v.78 (2009), 1647-1670.

http://tori.ils.uec.ac.jp/PAPERS/2010vladie.pdf D.Kouznetsov. Superexponential as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45.

Copyleft 2009–2011 by Dmitrii Kouznetsov: Use for free, attribute the source. The generators are available and will be uploaded upon request.

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