File:ExpQ2plotT.png

Explicit plot of exponential to base \(b\!=\!\sqrt{2} \approx 1.414213562373095\)

The thick curve: \(y=\exp_b(x)\).

The thin line shows the identical function, \(y\!=\!x\).

The fixed points \(L\!=\!2\) and \(L\!=\!4\) are solutions of the equation

\(\exp_b(L)=L\)

for \(b=\sqrt{2}\).

Any of these fixed points can be used to construct superexponentials to base \(\sqrt{2}~\) ^[1]. One of these superexponentials is qualified as Tetration to base sqrt2.

Using

The picture is used as Fig.9.1 at page 103 of book «Superfunctions» ^[2], 2020
and also at its preliminary (Russian) version «Суперфункции» ^[3], fig.9.1 at page 110.

References

C++ generator of curves

 #include<math.h>
 #include<stdio.h>
 #include<stdlib.h>
 #define DB double
 #define DO(x,y) for(x=0;x<y;x++)
 #include "ado.cin"

 DB B=sqrt(2.);

 main(){ int m,n; double x,y; FILE *o;
 o=fopen("ExpQ2plot.eps","w"); ado(o,1204,804);
 fprintf(o,"602 2 translate 100 100 scale\n");
 #define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
 #define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);
 for(m=-6;m<7;m++) {M(m,0)L(m,8)}
 for(m=0;m<9;m++) {M(-6,m)L(6,m)}
 fprintf(o,"2 setlinecap .01 W S\n 2 setlinecap 1 setlinejoin \n");
  for(m=0;m<123;m++){x=-6.1+.1*m; y=exp(log(B)*x); if(m==0)M(x,y) else L(x,y);} fprintf(o,".04 W .8 0 0 RGB S\n");
 M(-.1,-.1)L(6.1,6.1) fprintf(o,".016 W 0 0 0 RGB S\n\n");
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
   system("epstopdf ExpQ2plot.eps");
   system(    "open ExpQ2plot.pdf");            
   getchar(); system("killall Preview");//for mac
 }

Latex generator of labels

% file ExpQ2plot.pdf should be generated with the code above in order to compile the Latex document below.
% Copyleft 2012 by Dmitrii Kouznetsov

\documentclass[12pt]{article} 
\usepackage{geometry} 
\usepackage{graphicx} 
\usepackage{rotating} 
\paperwidth 1212pt 
\paperheight 844pt 
\topmargin -92pt 
\oddsidemargin -80pt 
\textwidth 1604pt 
\textheight 1604pt 
\pagestyle {empty} 
\newcommand \sx {\scalebox} 
\newcommand \rot {\begin{rotate}} 
\newcommand \ero {\end{rotate}} 
\newcommand \ing {\includegraphics} 
\parindent 0pt
\pagestyle{empty} 
\begin{document}
{\begin{picture}(1202,802)
\put(590,792){\sx{4.2}{$y$}}
\put(590,698){\sx{4.2}{$7$}} 
\put(590,598){\sx{4.2}{$6$}}
\put(590,498){\sx{4.2}{$5$}}
\put(590,398){\sx{4.2}{$4$}}
\put(590,298){\sx{4.2}{$3$}}
\put(590,198){\sx{4.2}{$2$}}
\put(590,098){\sx{4.2}{$1$}}
\put(080,-22){\sx{4}{$-5$}}
\put(180,-22){\sx{4}{$-4$}}
\put(281,-22){\sx{4}{$-3$}}
\put(381,-22){\sx{4}{$-2$}}
\put(482,-22){\sx{4}{$-\!1$}}
\put(603.6,-22){\sx{4}{$0$}}
\put(703.7,-22){\sx{4}{$1$}}
\put(803.8,-22){\sx{4}{$2$}}
\put(903.9,-22){\sx{4}{$3$}}
\put(1004.0,-22){\sx{4}{$4$}}
\put(1104.1,-22){\sx{4}{$5$}}
\put(1192.2,-22){\sx{4.3}{$x$}}
%\put(0815,520){\sx{5.6}{\rot{78}$y\!=\!\exp(x)$\ero}}
\put(1034,490){\sx{5.6}{\rot{66}$y\!=\!\exp_{_{\!\!\sqrt{2}}}(x)$\ero}}
\put(1094,444){\sx{5.7}{\rot{45}$y\!=\!x$\ero}}
\put(10,10){\ing{ExpQ2plot}}
\end{picture}}
\end{document} 

Keywords

«Base sqrt2», «[[]]», «Book», «Exponential», «Fixed point», «Superfunctions», «[[]]»,