File:Penplot.jpg: Difference between revisions

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== Summary ==
== Summary ==
{{Image_Details|user
Importing file
|description  = plot of the [[natural pension]] <math>\mathrm{pen}=\mathrm{pen}_{\mathrm e}</math>, id set, [[pentation]] to base <math>\mathrm e=\exp(1)\approx 2.71</math>, id set, [[pentation]] to base <math>\mathrm e=\exp(1)\approx 2.71</math>; the thik black curve shows <math>y=\mathrm{pen}(x)</math>; the thik black curve shows <math>y=\mathrm{pen}(x)</math>.
The thin curves show the two asymptotics of pentation and the error <math>\delta</math> of the linear approximation <math> z \mapsto 1+z</math>
|author      = [[User:Dmitrii Kouznetsov|Dmitrii Kouznetsov]]
|date-created = 2014
|pub-country  = Japan, Germany
|notes        =  [[Pentation]] is described at [[TORI]], http://mizugadro.mydns.jp/t/index.php/Pentation and also (In Russian) in the book <ref name="book">
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf  <br>
http://mizugadro.mydns.jp/BOOK/202.pdf
Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014. (In Russian), page 268, Figure 19.3.
</ref>
|versions    = The image is borrowed from [[TORI]], http://mizugadro.mydns.jp/t/index.php/File:Penplot.jpg
}}
 
== Licensing ==
{{CC|by|3.0}}
==Description==
[[Pentation]] pen is [[superfunction]] of [[tetration]] to the same base. Natural pentation is solution <math>F</math> of the [[transfer equation]]
 
<math>
F(z\!+\!1)=\mathrm{tet}\big( F(z)\big)
</math>
 
constructed with [[regular iteration]] at the smallest real [[fixed point]]  <math>L</math> of [[tetration]]; <math>L\approx -1.8503545290271812</math> is solution of equation
 
<math>L=\mathrm{tet}(L)</math>
 
with additional condition <math>F(0)=1</math>.
 
The real-real plot <math>y=\mathrm {pen}(x)</math> is shown with thick black curve.
 
The thin curves show approximations of pentation.
The red horizontal line shows the fixed point of tetration, <math>y=L</math>.
 
The thin blue curve shows the asymptotic of pentation at large negative values of the real part of the argument,
 
<math>
y= L+\exp(k(x+x_1))
</math>
 
where  <math>k\approx 1.86573322821</math>
 
and <math>x_1 \approx 2.24817451898</math>
 
The thin green line shown the deviation from the linear approximation
 
<math>\mathrm{linear}(x)=1+x</math>
 
The deviation is denoted as <math>~\delta(x)=\mathrm{pen}(x)-\mathrm{linear}(x)</math>
 
In the range <math>-2.1\!<\!x\!<\!1.1</math>, the deviation is small, the linear approximation provides 2 correct significant digits. In order to make the deviation visible, it is scaled with factor 10, so, <math>y=10\delta(x)</math> is plotted.
 
Properties of tetration are described in publications
<ref>
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. (2009). Solution of F(z+1)=exp(F(z)) in the complex z-plane. Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7.
</ref>
 
The [[regular iteration]] in construction of [[superfunction]] is described at [[TORI]], http://mizugadro.mydns.jp/t/index.php/Regular_iteration
and also in
<ref>
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.
</ref><ref>
http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
</ref><ref>
http://www.springerlink.com/content/u712vtp4122544x4 D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98.
http://tori.ils.uec.ac.jp/2012OR/2012or.pdf D. Kouznetsov. Superfunctions for optical amplifiers. Preprint ILS UEC, 2012
</ref>.
 
The picture is described in the book [[Суперфункции]], In Russian <ref name="book">
https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0 <br>
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf  <br>
http://mizugadro.mydns.jp/BOOK/202.pdf
Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014. (In Russian), page 268, Figure 19.3.
</ref>.
 
==[[C++]] generator of curves==
<nowiki>
 
#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 "ado.cin"
#include "fsexp.cin"
#include "fslog.cin"
 
z_type pen0(z_type z){
DB Lp=-1.8503545290271812;
DB k,a,b;
        k=1.86573322821; a=-.6263241; b=0.4827;
z_type e=exp(k*z);
return Lp + e*(1.+e*(a+b*e));
}
 
z_type pen7(z_type z){ DB x; int m,n; z=pen0(z+(2.24817451898-7.));
DO(n,7) { if(Re(z)>8.) return 999.; z=FSEXP(z); if(abs(z)<40) goto L1; return 999.; L1: ;}
return z; }
 
z_type pen(z_type z){ DB x; int m,n;
x=Re(z); if(x<= -4.) return pen0(z);
m=int(x+5.);
z-=DB(m);
z=pen0(z);
DO(n,m) z=FSEXP(z);
return z;
}
 
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
FILE *o;o=fopen("penplo.eps","w"); ado(o,608,1008);
fprintf(o,"404 204 translate\n 100 100 scale\n");
#define M(x,y) fprintf(o,"%8.4f %8.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%8.4f %8.4f L\n",0.+x,0.+y);
for(m=-4;m<3;m++) {M(m,-2)L(m,8)}
for(n=-2;n<11;n++) {M( -4,n)L(2,n)} fprintf(o,"2 setlinecap 1 setlinejoin .004 W 0 0 0 RGB S\n");
 
DO(n,150){x=-4+.04*n;y=Re(pen7(x)); if(n==0) M(x,y)else L(x,y); if(y>8.)break;} fprintf(o,".02 W 0 0 0 RGB S\n");
 
DO(n,150){x=-2.2+.04*n;y=10.*(Re(pen7(x))-(1.+x)); if(n==0) M(x,y)else L(x,y); if(y>.3)break;} fprintf(o,".01 W 0 .5 0 RGB S\n");
 
DB L=-1.8503545290271812;
DB K=1.86573322821;
DB a=-.6263241;
DB b=0.4827;
DO(n,80){x=-4.+.04*n; DB e=exp(K*(x+2.24817451898)); y=L+e;
        if(n==0) M(x,y) else L(x,y); if(y>8.) break;}
fprintf(o,".01 W 0 0 1 RGB S\n");
 
M(-4,L)L(0,L)
fprintf(o,".01 W 1 0 0 RGB S\n");
 
DB t2=M_PI/1.86573322821;
DB tx=-2.32;
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
 
printf("pen7(-1)=%18.14f\n", Re(pen7(-1.)));
printf("Pi/1.86573322821=%18.14f %18.14f\n", M_PI/1.86573322821, 2*M_PI/1.86573322821);
 
        system("epstopdf penplo.eps");
        system( "open penplo.pdf");
}
</nowiki>
 
==[[Latex]] generator of labels==
<nowiki>
 
\documentclass[12pt]{article}
\paperwidth 608px
\paperheight 1008px
\textwidth 1394px
\textheight 1300px
\topmargin -104px
\oddsidemargin -90px
\usepackage{graphics}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\newcommand \rmi {\mathrm{i}}
\begin{document}
{\begin{picture}(608,1008)
\put(0,0){\ing{penplo}}
\put(377,994){\sx{3.2}{$y$}}
\put(377,895){\sx{3.2}{$7$}}
\put(377,795){\sx{3.2}{$6$}}
\put(377,695){\sx{3.2}{$5$}}
\put(377,594){\sx{3.2}{$4$}}
\put(377,494){\sx{3.2}{$3$}}
\put(377,394){\sx{3.2}{$2$}}
\put(377,294){\sx{3.2}{$1$}}
\put(377,194){\sx{3.2}{$0$}}
\put(358, 93){\sx{3.2}{$-1$}}
\put(80,174){\sx{3.2}{$-3$}}
\put(180,174){\sx{3.2}{$-2$}}
\put(280,174){\sx{3.2}{$-1$}}
\put(396,174){\sx{3.2}{$0$}}
\put(496,174){\sx{3.2}{$1$}}
\put(590,174){\sx{3.2}{$x$}}
\put(242,406){\sx{3.6}{\rot{85}$y\!=\!L+\exp(k(x\!+\!x_1))$\ero}}
\put(446,370){\sx{3.9}{\rot{70}$y\!=\!\mathrm{pen}(x)$\ero}}
\put(8,236){\sx{3.3}{$y=10\,\delta(x)$}}
\put(312, 9){\sx{3.2}{$y\!=\!L$}}
\end{picture}
\end{document}
</nowiki>
==References==
<references/>
 
[[Category:Tetration]]
[[Category:Pentation]]
[[Category:Superfunction]]
[[Category:Explicit plot]]
[[Category:TORI]]

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