File:IterEq2plotU.png

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 , these F and G are referred as $F_{4,5}$ and $F_{4,5}^{~-1}$, respectively.

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:

using namespace std; typedef complex z_type;
 * 1) include
 * 2) include
 * 3) include
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "ado.cin"


 * 1) include "F45E.cin"
 * 2) 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"); 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"); }
 * 1) define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);

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} % %