Difference between revisions of "File:Sqrt2sufuplot.png"

Four superexponentials to base \(b=\sqrt{2}\)

They are real-holomorphic solutions \(F\) of the transfer equation

\( \exp_b(F(z))=F(z\!+\!1) \)

with different values at zero and different asymptotic properties.

Along the real axis, functions \(F_{2,3} \) and \(F_{4,3} \) coincide with 24 decimal figures; the deviation

\( d_{42}(x)=F_{4,3}(x)-F_{2,3}(x) \)

scaled with factor \(10^{24} \) is shown with dark pink oscillating curve.

Usage

This image is used in book Суперфункции (2014, In Russian) ^[1] is figure 16.10, page 237.

This image is used in book Superfunctions (2020, in English) ^[2] as figure 16.9, page 232.

The English version ^[2] happen to be better, than the Russian ^[1] one; so, ^[2] can be recommended for the description.

This image is used also in article with Henryk Trappmann ^[3], Fig.3, page 1740.
Henryk had asked the Editor to consider superexponentials to base \( b \!=\!\sqrt{2} \); this pic had been plotted to satisfy his curiosity.

See the description in the cited literature. The description happens to be a little bit long; so, it is not reproduced here).

C++ generator of the curves

/* Files ado.cin, conto.cin, difapro.cin, sqrt2f21e.cin, sqrt2f23e.cin, sqrt2f43e.cin, sqrt2f45e.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 "ado.cin"
#include "sqrt2f21e.cin"
#include "sqrt2f23e.cin"
#include "sqrt2f43e.cin"
#include "sqrt2f45e.cin"
//#include "superex.cin"
//#include "slog14128.cin" 
#include "difapro.cin"
int  main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
//  FILE *o;o=fopen("rearea1.eps","w");  ado(o,202,148);
  FILE *o;o=fopen("sqrt23.eps","w");  ado(o,202,148);
 fprintf(o,"101 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(0,10.06)
M(-10.06,0)L(10.06,0)
fprintf(o,".03 W S\n");
for(n=-10;n<11;n++){if(n!=0){M(n,0)L(n,-.1)}}
for(n= -4;n<11;n++){if(n!=0){M(0,n)L(-.1,n)}}
fprintf(o,".03 W S\n");
M(-2,-4)L(-2,0)
M(0,2)L(10,2)
//M(-2,2)L(4,-4)
M(-10,4)L(0,4)
fprintf(o,".01 W S\n");

fprintf(o,"1 setlinejoin 2 setlinecap\n");
for(m=0;m< 84;    m+=4) { x=-1.84+.01*m; z=x;  y=Re(F21E(z)); if(m==0) M(x,y) else L(x,y)}
for(m=0;m<511;   m+=10){ x=-1.  +.01*m; z=x;  y=Re(F21E(z)); L(x,y)}
for(m=520;m<1101;m+=20){ x=-1.  +.01*m; z=x;  y=Re(F21E(z)); L(x,y)}
fprintf(o,".05 W 0 0 1 RGB S\n");

for(m=0;m<201;m+=5) { x=-10+.1*m; z=x;  y=Re(F23E(z)); if(m==0) M(x,y) else L(x,y)}
fprintf(o,".05 W 0 .8 0 RGB S\n");

//for(m=0;m<201;m+=5) { x=-10+.1*m; z=x;  y=Re(F43E(z)); if(m==0) M(x,y) else L(x,y)}
//fprintf(o,".02 W .5 0 .5 RGB S\n");

for(m=0;m<166;m+=5) { x=-10+sqrt(1.*m); z=x;  y=Re(F45E(z)); if(m==0) M(x,y) else L(x,y)}
fprintf(o,".03 W 1 0 0 RGB S\n");

//for(m=0;m<386;m+=5) { x=-1.984+.01*m; z=x;  y=Re(FSEXP(z)); if(m==0) M(x,y) else L(x,y)}
//fprintf(o,".07 W 0 1 1 RGB S\n");

//for(m=0;m<145;m+=2) { y=-4+.1*m; z=z_type(y,0.); x=Re(FSLOG(z)); if(m==0) M(x,y) else L(x,y)}
//fprintf(o,".012 W 0 0 0 RGB S\n");

DO(m,502){      x=-10.+.04*m; 
                z=z_type(x,0.);
                c=difapro(z);
                y=Re(c)*1.e24;
                if(m==0) M(x,y) else L(x,y)}
fprintf(o,".02 W .6 0 .6 RGB S\n");


 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf sqrt23.eps"); 
        system(    "open sqrt23.pdf");
        getchar(); system("killall Preview"); // For macintosh
}

Latex generator of the labels

\documentclass[12pt]{article} 
\usepackage{geometry} 
\usepackage{graphicx} 
\usepackage{rotating} 
\paperwidth 424pt 
\paperheight 302pt 
\topmargin -106pt 
\oddsidemargin -73pt 
\textwidth 1064pt 
\textheight 1060pt 
\pagestyle {empty} 
\newcommand \sx {\scalebox} 
\newcommand \rot {\begin{rotate}} 
\newcommand \ero {\end{rotate}} 
\newcommand \ing {\includegraphics} 
\parindent 0pt
\pagestyle{empty} 
\begin{document} \sx{2.1}{\begin{picture}(146,142) 
%\put(10,10){\ing{IterPowPlot}} 
%\put(40,40){\ing{Itereq2tlo}} 
\put(0,0){\ing{sqrt23}} 
\put(95,139){\sx{.7}{$y$}}
\put(95,119){\sx{.6}{$8$}}
\put(95, 99){\sx{.6}{$6$}}
\put(95, 79){\sx{.6}{$4$}}
\put(95, 59){\sx{.6}{$2$}}
\put(91, 19){\sx{.6}{$-2$}}
\put( 17,35){\sx{.6}{$-\!8$}}
\put( 37,35){\sx{.6}{$-\!6$}}
\put( 57,35){\sx{.6}{$-\!4$}}
\put( 77,35){\sx{.6}{$-\!2$}}
\put(120,35){\sx{.6}{$2$}}
\put(140,35){\sx{.6}{$4$}}
\put(160,35){\sx{.6}{$6$}}
\put(180,35){\sx{.6}{$8$}}
\put(197.6,35){\sx{.7}{$x$}}
%
\put(110,93){\sx{.6}{$y\!=\!F_{4,5}(x)\!=\!\mathrm{SuExp}_{\sqrt{2},5}(x)$}}
%\put(116, 88){\sx{.7}{$y\!=\!\mathrm{tet}(x)$}}
\put(126, 69){\sx{.6}{$y\!=\!F_{2,3}(x)$,  $y\!=\!F_{4,3}(x)$}}
\put(126, 54){\sx{.6}{$y\!=\!F_{2,1}(x)\!=\!\mathrm{tet}_{\sqrt{2}}(x)$}}
%\put(108, 44){\sx{.7}{$y\!=\!10^{24}w(x)$}}
%\enp}
\put(126,44){\sx{.6}{$y\!=\!10^{24} d_{42}(x)$}}
\end{picture}}
\end{document} 

References

Keywords

«[[]]», «Base sqrt2», «Henryk Trappmann», «Superexponential», «Superfunction», «Superfunctions», «Tetration»,

«Суперфункции»,