File:IterEq2plotU.png

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Explicit plot of $c$th iteration of exponential to base sqrt(2) for various values of the number $c$ of iterations.

For evaluation of the non-integer iteration, the plotter uses the implementation through the superfunction $F$ of the exponential to base $\sqrt{2}$, constructed at the fixed point $L\!=\!4$, and the corresponding Abel function $G$:

$ \exp_b^{c}(x)=F\big(c+G(x)\big)$

Note: In publication ^[1], these F and G are referred as $F_{4,5}$ and $F_{4,5}^{~-1}$, respectively.

References

↑ http://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf D.Kouznetsov, H.Trappmnn. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, v.271, July 2010, p.1727-1756.

C++ generator of curves

// Files F45E.cin, F45L.cin and ado.cin should be loaded in the working directory in order to compile the C++ code below:

#include<math.h>
#include<stdio.h>
#include<stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
using namespace std;
#include <complex>
typedef complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "ado.cin"

#include "F45E.cin"
#include "F45L.cin"

DB B=sqrt(2.);
DB F(DB z) { return exp( exp( log(B)*z));}
DB G(DB z) { return log( log(z) )/log(B);}

main(){ int m,n; double x,y,t; FILE *o;
o=fopen("IterEq2plot.eps","w"); ado(o,1420,1420);
fprintf(o,"1 1 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=0;m<15;m++) {M(m,0)L(m,14)}
for(m=0;m<15;m++) {M(0,m)L(14,m)}
fprintf(o,"2 setlinecap .01 W S\n");
DO(m,82){x=0.001+.1*m; y=exp(log(B)*x); y=exp(log(B)*y); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1  setlinejoin .04 W 0 1 0 RGB S\n");
DO(m,82){x=0.001+.1*m; y=exp(log(B)*x); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1  setlinejoin .04 W 0 1 0 RGB S\n");
DO(m,141){x=0.001+.1*m; y=log(x)/log(B); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 1 0 1 RGB S\n");
DO(m,131){x=1.421+.1*m;y=log(x)/log(B);y=log(y)/log(B); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 1 0 1 RGB S\n");
for(n=-10;n<11;n++){t=.1*n; M(2,2); DO(m,122){x=2.05+.1*m; y=Re(F45E(t+F45L(x+1.e-14*I))); L(x,y); if(y>14.1)break;} }
fprintf(o,"1 setlinecap 1 setlinejoin .02 W 0 0 0 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
      system("epstopdf IterEq2plot.eps");    
      system(    "open IterEq2plot.pdf");
      getchar(); system("killall Preview");
}

Latex generator of labels

%

% 
% file IterEq2plot.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 1418pt % 
\paperheight 1418pt % 
\topmargin -94pt % 
\oddsidemargin -81pt % 
\textwidth 1600pt % 
\textheight 1600pt % 
\pagestyle {empty} % 
\newcommand \sx {\scalebox} % 
\newcommand \rot {\begin{rotate}} % 
\newcommand \ero {\end{rotate}} % 
\newcommand \ing {\includegraphics} % 
\parindent 0pt% 
\pagestyle{empty} % 
\begin{document} % 
\begin{picture}(1412,1412) % 
%\put(10,10){\ing{IterPowPlot}} % 
\put(10,10){\ing{IterEq2plot}} % 
\put(11,1374){\sx{4.7}{$y\!=\!\exp_{b}^{~ c}(x)$}} % 
\put(11,1298){\sx{4}{$13$}} % 
\put(11,1198){\sx{4}{$12$}} % 
\put(11,1098){\sx{4}{$11$}} % 
\put(11,998){\sx{4}{$10$}} % 
\put(11,898){\sx{4}{$9$}} % 
\put(11,798){\sx{4}{$8$}} % 
\put(11,698){\sx{4}{$7$}} % 
\put(11,598){\sx{4}{$6$}} % 
\put(11,498){\sx{4}{$5$}} % 
\put(11,398){\sx{4}{$4$}} % 
\put(11,298){\sx{4}{$3$}} % 
\put(11,198){\sx{4}{$2$}} % 
\put(11,098){\sx{4}{$1$}} % 
% 
\put(100,16){\sx{4}{$1$}} % 
\put(200,16){\sx{4}{$2$}} % 
\put(301,16){\sx{4}{$3$}} % 
\put(401,16){\sx{4}{$4$}} % 
\put(502,16){\sx{4}{$5$}} % 
\put(602,16){\sx{4}{$6$}} % 
\put(703,16){\sx{4}{$7$}} % 
\put(803,16){\sx{4}{$8$}} % 
\put(903,16){\sx{4}{$9$}} % 
\put(993,16){\sx{4}{$10$}} % 
\put(1093,16){\sx{4}{$11$}} % 
\put(1193,16){\sx{4}{$12$}} % 
\put(1293,16){\sx{4}{$13$}} % 
\put(1390,16){\sx{4}{$x$}} % 
% 
\put(595,1250){\sx{5}{\rot{84}$c\!=\!2$\ero}} % 
\put(754,1250){\sx{5}{\rot{78}$c\!=\!1$\ero}} % 
\put(809,1250){\sx{5}{\rot{74}$c\!=\!0.8$\ero}} % 
\put(882,1250){\sx{5}{\rot{72}$c\!=\!0.6$\ero}} % 
\put(971,1250){\sx{5}{\rot{64}$c\!=\!0.4$\ero}} % 
\put(1094,1250){\sx{5}{\rot{54}$c\!=\!0.2$\ero}} % 
% 
\put(1282,1262){\sx{5}{\rot{44}$c\!=\!0$\ero}} % 
% 
\put(1256,1060){\sx{5}{\rot{34}$c\!=\!-0.2$\ero}} % 
\put(1235, 926){\sx{5}{\rot{26}$c\!=\!-0.4$\ero}} % 
\put(1230, 840){\sx{5}{\rot{19}$c\!=\!-0.6$\ero}} % 
\put(1220, 770){\sx{5}{\rot{15}$c\!=\!-0.8$\ero}} % 
\put(1230, 702){\sx{5}{\rot{11}$c\!=\!-1$\ero}} % 
\put(1234, 542){\sx{5}{\rot{5}$c\!=\!-2$\ero}} % 
% 
\put(560, 1032){\sx{5.4}{\rot{83}$y\!=\!b^{b^x}$\ero}} % 
\put(674, 1062){\sx{5}{\rot{72}$y\!=\!b^x$\ero}} % 
\put(890,610){\sx{5}{\rot{12}$y\!=\!\log_b(x)$\ero}} % 
\put(825,484){\sx{5}{\rot{6}$y\!=\!\log_b^{~2}(x)$\ero}} % 
\put(600,200){\sx{11}{$b\!=\!\sqrt{2}$}}
\end{picture} % 
\end{document} % 
%

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	Date/Time	Thumbnail	Dimensions	User	Comment
current	17:50, 20 June 2013		2,944 × 2,944 (986 KB)	Maintenance script (talk \| contribs)	Importing image file

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[1] ttp://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf D.Kouznetsov, H.Trappmnn. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, v.271, July 2010, p.1727-1756.

[1]