Difference between revisions of "File:Sqrt27u.png"

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Check of the approximate symmetry of the [[explicit plot]] of [[tetration]] to base $\sqrt{2}$ in figure http://mizugadro.mydns.jp/t/index.php/File:Sqrt27t.jpg
−
Importing image file
  
  +
  +
$y=\mathrm{devi}(x)=\mathrm{tet}_{\sqrt{2}}(x) + \mathrm{ate}_{\sqrt{2}}(-x) ~$
  +
(dashed line) and
  +
  +
$y=\mathrm{tet}_{\sqrt{2}}(-\mathrm{ate}_{\sqrt{2}}(x))+x ~$
  +
(solid line)
  +
  +
  +
Usage: this is figure 16.6 of the book [[Суперфункции]] (2014, In Russian) <ref>
  +
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.
  +
</ref>; the English version is in preparation in 2015.
  +
  +
This image is used also as figure 7b in the article
  +
<ref>
  +
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html <br>
  +
http://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf offprint
  +
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>.
  +
  +
  +
==Refereces==
  +
<references/>
  +
  +
  +
==[[C++]] generator of the curves==
  +
Files [[ado.cin]],
  +
[[sqrt2f21e.cin]],
  +
[[sqrt2f21l.cin]]
  +
should be loaded in order to compile the code below.
  +
  +
<poem><nomathjax><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 "conto.cin"
  +
#include "ado.cin"
  +
//#include "tq2e.cin"
  +
#include "sqrt2f21e.cin"
  +
#include "sqrt2f21l.cin"
  +
  +
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  +
FILE *o;o=fopen("sqrt27b.eps","w"); ado(o,104,63);
  +
fprintf(o,"21 41 translate\n 10 10 scale\n");
  +
#define M(x,y) fprintf(o,"%7.4f %7.4f M\n",0.+x,0.+y);
  +
#define L(x,y) fprintf(o,"%7.4f %7.4f L\n",0.+x,0.+y);
  +
M(0,-4.06)L(0,2.06)
  +
M(-2.06,0)L(8.06,0)
  +
fprintf(o,".014 W S\n");
  +
for(n=-2;n<9;n++){if(n!=0){M(n,0)L(n,-.1)}}
  +
for(n=-4;n<3;n++){if(n!=0){M(0,n)L(-.1,n)}}
  +
fprintf(o,".02 W S\n");
  +
  +
/*
  +
M(-2,-4)L(-2,2)L(4,2)
  +
M(-2,2)L(4,-4)
  +
*/
  +
  +
M(-2,-4)L(-2,2) fprintf(o,".006 W S\n");
  +
  +
fprintf(o,"1 setlinejoin 0 setlinecap\n");
  +
  +
for(m=0;m< 99;m+= 2){x=-1.9999+.01*m; z=x; c=F21E(-F21E(z))+z; y=Re(c);y*=100; if(m==0) M(x,y) else L(x,y)}
  +
for(m=0;m<381;m+=10){x=-1.+.01*m; z=x; c=F21E(-F21E(z))+z; y=Re(c);y*=100; L(x,y)}
  +
fprintf(o,".01 W 1 0 0 RGB S\n");
  +
  +
  +
//for(m=0;m< 99;m+= 2){x=-1.8+.01*m; z=x;c=F21E(z)+F21L(-z); y=Re(c);y*=100; if(m==0) M(x,y) else L(x,y)}
  +
//for(m=0;m<381;m+=10){x=-1.+.01*m; z=x; c=F21E(z)+F21L(-z); y=Re(c);y*=100; L(x,y)}
  +
  +
//for(m=0;m<101;m+=2){x=-1.738+(.0012*m*m); z=x;c=F21E(z)+F21L(-z); y=Re(c);y*=100; if(m==0) M(x,y) else L(x,y)}
  +
for(m=0;m<101;m+=2){x=-1.738+(.00097*m*(m+2)); z=x;c=F21E(z)+F21L(-z); y=Re(c);y*=100; if(m==0) M(x,y) else L(x,y)}
  +
fprintf(o,"[.1 .1] 0 setdash .03 W 0 .8 0 RGB S\n");
  +
  +
/*
  +
for(m=0;m<101;m+=5){x=-1.85+(.0012*m*m); z=x;c=F21L(-z)/F21E(z)+1.; y=Re(c);y*=100; if(m==0) M(x,y) else L(x,y)}
  +
fprintf(o,"[.1 .1] 0 setdash .02 W 0 0 1 RGB S\n");
  +
  +
for(m=0;m<102;m+=2){x=-1.998+(.0012*m*m); z=x;
  +
c=F21E(z); d=-F21L(-z); c=(c-d)/(c+d);
  +
y=Re(c); y*=100.; if(m==0) M(x,y) else L(x,y)}
  +
fprintf(o,"[.05 .05] 0 setdash .006 W 0 0 0 RGB S\n");
  +
*/
  +
// c=TQ2E(z);
  +
// p=Re(c);
  +
// #include "plof.cin"
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf sqrt27b.eps");
  +
system( "open sqrt27b.pdf"); //for LINUX
  +
// getchar(); system("killall Preview"); // For macintosh
  +
}
  +
  +
</nowiki></nomathjax></poem>
  +
  +
==[[Latex]] generator of the labels==
  +
  +
<poem><nomathjax><nowiki>
  +
\documentclass[12pt,a4paper,oneside]{book}
  +
%\newcommand \EN[1] {{#1}} % make the English version
  +
\newcommand \EN[1] {{}} % suppress the English version
  +
\newcommand \RU[1] {{#1}} % make the Russian version (in this document not supported)
  +
%\newcommand \RU[1] {{}} % suppress the Russian version
  +
%The Japanese version is not yet supported. While \JP is used to suppress several lines at once.
  +
\newcommand \JP[1] {{}} % ореsuppress some text
  +
%\usepackage[space]{cite }% If exist.
  +
\usepackage[utf8]{inputenc}
  +
\usepackage[T2A]{fontenc}
  +
\usepackage[russian]{babel}
  +
\usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx}
  +
  +
\usepackage{rotating}
  +
\usepackage{hyperref}
  +
\usepackage{wrapfig}
  +
\usepackage{geometry}
  +
\paperwidth 145mm
  +
\paperheight 86mm
  +
\textwidth 550mm
  +
%\oddsidemargin -76pt
  +
\oddsidemargin -70pt
  +
\topmargin -97pt
  +
\textheight 457mm
  +
\pagestyle{empty}
  +
  +
\newcommand \sx {\scalebox}
  +
\newcommand \rot {\begin{rotate}}
  +
\newcommand \ero {\end{rotate}}
  +
\newcommand \ing {\includegraphics}
  +
\parindent 0pt
  +
\begin{document}
  +
\sx{4}{\begin{picture}(100,60)
  +
\put(0,0){\ing{sqrt27b}}
  +
\put(17,58.8){\sx{.36}{$y$}}
  +
%\put(60,48){\sx{.4}{$y=F_{2,1}(x)+F_{2,1}^{-1}(\!-x)$}}
  +
%\put(48, 8){\sx{.4}{$y=F_{2,1}(-F_{2,1}(x))+x$}}
  +
\put(46,48){\sx{.4}{$y=\mathrm{tet}_{\sqrt{2}}(x)+\mathrm{ate}_{\sqrt{2}}(\!-x)$}}
  +
\put(47, 8){\sx{.4}{$y=\mathrm{tet}_{\sqrt{2}}(-\mathrm{tet}_{\sqrt{2}}(x))+x$}}
  +
%\put(184,600){\sx{3.3}{$2$}}
  +
\put(12.3,50.0){\sx{.33}{$0.01$}}
  +
\put(10.0,30.0){\sx{.33}{$-0.01$}}
  +
\put(10.0,20.0){\sx{.33}{$-0.02$}}
  +
\put(10.0,10.0){\sx{.33}{$-0.03$}}
  +
%\put(160, 0){\sx{3.3}{$-4$}}
  +
\put( -1,37.4){\sx{.33}{$-2$}}
  +
\put( 9.0,37.4){\sx{.33}{$-1$}}
  +
\put(30.2,37.4){\sx{.33}{$1$}}
  +
\put(40.3,37.4){\sx{.33}{$2$}}
  +
\put(50.3,37.4){\sx{.33}{$3$}}
  +
\put(60.3,37.4){\sx{.33}{$4$}}
  +
\put(70.4,37.4){\sx{.33}{$5$}}
  +
\put(80.4,37.4){\sx{.33}{$6$}}
  +
\put(90.4,37.4){\sx{.33}{$7$}}
  +
\put(100.,37.6){\sx{.36}{$x$}}
  +
%\put(603,377){\sx{3.4}{$4$}}
  +
%\put(60.0,42.0){\sx{.42}{$x$}}
  +
\end{picture}}
  +
\end{document}
  +
  +
</nowiki></nomathjax></poem>
  +
  +
[[Category:Base sqrt2]]
  +
[[Category:Sqrt2]]
  +
[[Category:Book]]
  +
[[Category:BookPlot]]
  +
[[Category:C++]]
  +
[[Category:Explicit plot]]
  +
[[Category:Exp]]
  +
[[Category:Generator]]
  +
[[Category:Latex]]
  +
[[Category:Superfunction]]
  +
[[Category:Tetration]]

Latest revision as of 08:52, 1 December 2018

Check of the approximate symmetry of the explicit plot of tetration to base $\sqrt{2}$ in figure http://mizugadro.mydns.jp/t/index.php/File:Sqrt27t.jpg

$y=\mathrm{devi}(x)=\mathrm{tet}_{\sqrt{2}}(x) + \mathrm{ate}_{\sqrt{2}}(-x) ~$ (dashed line) and

$y=\mathrm{tet}_{\sqrt{2}}(-\mathrm{ate}_{\sqrt{2}}(x))+x ~$ (solid line)


Usage: this is figure 16.6 of the book Суперфункции (2014, In Russian) [1]; the English version is in preparation in 2015.

This image is used also as figure 7b in the article [2].


Refereces

  1. ↑ https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
    http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
    http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Publishing, 2014.
  2. ↑ http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html
    http://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf offprint D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-1756.


C++ generator of the curves

Files ado.cin, sqrt2f21e.cin, sqrt2f21l.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"
#include "ado.cin"
//#include "tq2e.cin"
#include "sqrt2f21e.cin"
#include "sqrt2f21l.cin"

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  FILE *o;o=fopen("sqrt27b.eps","w"); ado(o,104,63);
 fprintf(o,"21 41 translate\n 10 10 scale\n");
#define M(x,y) fprintf(o,"%7.4f %7.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%7.4f %7.4f L\n",0.+x,0.+y);
M(0,-4.06)L(0,2.06)
M(-2.06,0)L(8.06,0)
fprintf(o,".014 W S\n");
for(n=-2;n<9;n++){if(n!=0){M(n,0)L(n,-.1)}}
for(n=-4;n<3;n++){if(n!=0){M(0,n)L(-.1,n)}}
fprintf(o,".02 W S\n");

/*
M(-2,-4)L(-2,2)L(4,2)
M(-2,2)L(4,-4)
*/

M(-2,-4)L(-2,2) fprintf(o,".006 W S\n");

fprintf(o,"1 setlinejoin 0 setlinecap\n");

for(m=0;m< 99;m+= 2){x=-1.9999+.01*m; z=x; c=F21E(-F21E(z))+z; y=Re(c);y*=100; if(m==0) M(x,y) else L(x,y)}
for(m=0;m<381;m+=10){x=-1.+.01*m; z=x; c=F21E(-F21E(z))+z; y=Re(c);y*=100; L(x,y)}
fprintf(o,".01 W 1 0 0 RGB S\n");


//for(m=0;m< 99;m+= 2){x=-1.8+.01*m; z=x;c=F21E(z)+F21L(-z); y=Re(c);y*=100; if(m==0) M(x,y) else L(x,y)}
//for(m=0;m<381;m+=10){x=-1.+.01*m; z=x; c=F21E(z)+F21L(-z); y=Re(c);y*=100; L(x,y)}

//for(m=0;m<101;m+=2){x=-1.738+(.0012*m*m); z=x;c=F21E(z)+F21L(-z); y=Re(c);y*=100; if(m==0) M(x,y) else L(x,y)}
  for(m=0;m<101;m+=2){x=-1.738+(.00097*m*(m+2)); z=x;c=F21E(z)+F21L(-z); y=Re(c);y*=100; if(m==0) M(x,y) else L(x,y)}
fprintf(o,"[.1 .1] 0 setdash .03 W 0 .8 0 RGB S\n");

/*
for(m=0;m<101;m+=5){x=-1.85+(.0012*m*m); z=x;c=F21L(-z)/F21E(z)+1.; y=Re(c);y*=100; if(m==0) M(x,y) else L(x,y)}
fprintf(o,"[.1 .1] 0 setdash .02 W 0 0 1 RGB S\n");

for(m=0;m<102;m+=2){x=-1.998+(.0012*m*m); z=x;
                c=F21E(z); d=-F21L(-z); c=(c-d)/(c+d);
                y=Re(c); y*=100.; if(m==0) M(x,y) else L(x,y)}
fprintf(o,"[.05 .05] 0 setdash .006 W 0 0 0 RGB S\n");
*/
 // c=TQ2E(z);
 // p=Re(c);
// #include "plof.cin"
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf sqrt27b.eps");
        system( "open sqrt27b.pdf"); //for LINUX
// getchar(); system("killall Preview"); // For macintosh
}

Latex generator of the labels


\documentclass[12pt,a4paper,oneside]{book}
%\newcommand \EN[1] {{#1}} % make the English version
\newcommand \EN[1] {{}} % suppress the English version
\newcommand \RU[1] {{#1}} % make the Russian version (in this document not supported)
%\newcommand \RU[1] {{}} % suppress the Russian version
%The Japanese version is not yet supported. While \JP is used to suppress several lines at once.
\newcommand \JP[1] {{}} % ореsuppress some text
%\usepackage[space]{cite }% If exist.
\usepackage[utf8]{inputenc}
\usepackage[T2A]{fontenc}
\usepackage[russian]{babel}
\usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx}

\usepackage{rotating}
\usepackage{hyperref}
\usepackage{wrapfig}
\usepackage{geometry}
\paperwidth 145mm
\paperheight 86mm
\textwidth 550mm
%\oddsidemargin -76pt
\oddsidemargin -70pt
\topmargin -97pt
\textheight 457mm
\pagestyle{empty}

\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\parindent 0pt
\begin{document}
\sx{4}{\begin{picture}(100,60)
\put(0,0){\ing{sqrt27b}}
\put(17,58.8){\sx{.36}{$y$}}
%\put(60,48){\sx{.4}{$y=F_{2,1}(x)+F_{2,1}^{-1}(\!-x)$}}
%\put(48, 8){\sx{.4}{$y=F_{2,1}(-F_{2,1}(x))+x$}}
\put(46,48){\sx{.4}{$y=\mathrm{tet}_{\sqrt{2}}(x)+\mathrm{ate}_{\sqrt{2}}(\!-x)$}}
\put(47, 8){\sx{.4}{$y=\mathrm{tet}_{\sqrt{2}}(-\mathrm{tet}_{\sqrt{2}}(x))+x$}}
%\put(184,600){\sx{3.3}{$2$}}
\put(12.3,50.0){\sx{.33}{$0.01$}}
\put(10.0,30.0){\sx{.33}{$-0.01$}}
\put(10.0,20.0){\sx{.33}{$-0.02$}}
\put(10.0,10.0){\sx{.33}{$-0.03$}}
%\put(160, 0){\sx{3.3}{$-4$}}
\put( -1,37.4){\sx{.33}{$-2$}}
\put( 9.0,37.4){\sx{.33}{$-1$}}
\put(30.2,37.4){\sx{.33}{$1$}}
\put(40.3,37.4){\sx{.33}{$2$}}
\put(50.3,37.4){\sx{.33}{$3$}}
\put(60.3,37.4){\sx{.33}{$4$}}
\put(70.4,37.4){\sx{.33}{$5$}}
\put(80.4,37.4){\sx{.33}{$6$}}
\put(90.4,37.4){\sx{.33}{$7$}}
\put(100.,37.6){\sx{.36}{$x$}}
%\put(603,377){\sx{3.4}{$4$}}
%\put(60.0,42.0){\sx{.42}{$x$}}
\end{picture}}
\end{document}

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