Difference between revisions of "File:Sqrt27u.png"

Fig.16.5 from page 225 of book «Superfunctions» ^[1], 2020.

This image is also used as Рис.16.6 at page 230 of the Russian version «Суперфункции» ^[2], 2014.

This plot is used also as figure 7b in the article ^[3] at Mathematics of Computation, 2010.

The figure shows the numerical 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

The following curves are drawn:

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

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

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}

References

Keywords

«Arctetration», «Base sqrt2», «Sqrt2», «Exp», «Exponential», «Superfunction», «Superfunctions», «Tetration»,

«ado.cin», «sqrt2f21e.cin», «sqrt2f21l.cin»,

«Тетрация», «Суперфункции»,