File:IterEq2plotT.jpg

Explicit plot of $n$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^{n}(x)=F\big(n+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);}

int 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");
 * 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);

M(0,1.99)L(3.995,2.01)L(4.02,24) fprintf(o,"1 setlinecap 1 setlinejoin .03 W 0 .8 0 RGB S\n"); M(1.99,0)L(2.01,3.995)L(14,4.02) fprintf(o,"1 setlinecap 1 setlinejoin .03 W .8 0 .8 RGB S\n"); M(0,0)L(14,14) fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 0 1 RGB S\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 0 0 0 RGB S\n"); DO(m,82){x=0.001+.1*m;y=exp(log(B)*x); y=exp(log(B)*y); y=exp(log(B)*y); y=exp(log(B)*y); if(m==0)M(x,y) else L(x,y); if(y>15.1) break;} DO(m,82){x=0.001+.1*m; y=exp(log(B)*x);y=exp(log(B)*y); y=exp(log(B)*y); if(m==0) M(x,y) else L(x,y);if(y>15.1) break;} 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); 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 .9 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 .9 0 .9 RGB S\n"); DO(m,131){x=1.41+.1*m;y=log(x)/log(B);y=log(y)/log(B); if(m==0)M(x,y) else L(x,y);} DO(m,131){x=1.63+.1*m;y=log(x)/log(B);y=log(y)/log(B);y=log(y)/log(B); if(m==0)M(x,y) else L(x,y);} DO(m,131){x=1.75+.1*m;y=log(x)/log(B);y=log(y)/log(B);y=log(y)/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=-20;n<21;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
\documentclass[12pt]{article} \usepackage{geometry} \usepackage{graphicx} \usepackage{rotating} \paperwidth 1408pt \paperheight 1408pt \topmargin -103pt \oddsidemargin -73pt \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}(1402,1402) %\put(10,10){\ing{IterPowPlot}} \put(1,1){\ing{IterEq2plot}} %\put(0,0){\ing{ZexIte}} \put(11,1384){\sx{4.4}{$y$}} \put(04,1290){\sx{4}{$13$}} \put(04,1190){\sx{4}{$12$}} \put(04,1090){\sx{4}{$11$}} \put(04,990){\sx{4}{$10$}} \put(11,890){\sx{4}{$9$}} \put(11,790){\sx{4}{$8$}} \put(11,690){\sx{4}{$7$}} \put(11,590){\sx{4}{$6$}} \put(11,490){\sx{4}{$5$}} \put(11,390){\sx{4}{$4$}} \put(11,290){\sx{4}{$3$}} \put(11,190){\sx{4}{$2$}} \put(11,090){\sx{4}{$1$}} \put(91,6){\sx{4}{$1$}} \put(191,6){\sx{4}{$2$}} \put(291,6){\sx{4}{$3$}} \put(391,6){\sx{4}{$4$}} \put(492,6){\sx{4}{$5$}} \put(592,6){\sx{4}{$6$}} \put(693,6){\sx{4}{$7$}} \put(794,6){\sx{4}{$8$}} \put(894,6){\sx{4}{$9$}} \put(982,6){\sx{4}{$10$}} \put(1082,6){\sx{4}{$11$}} \put(1182,6){\sx{4}{$12$}} \put(1282,6){\sx{4}{$13$}} \put(1380,6){\sx{4.4}{$x$}}

\put(416,1158){\sx{5}{\rot{90}$n\!\rightarrow \! \infty$\ero}} \put(518,1250){\sx{5}{\rot{88}$n\!=\!3$\ero}} \put(590,1250){\sx{5}{\rot{84}$n\!=\!2$\ero}} % \put(750,1250){\sx{5}{\rot{78}$n\!=\!1$\ero}} % \put(800,1240){\sx{5}{\rot{74}$n\!=\!0.8$\ero}} % \put(872,1240){\sx{5}{\rot{71}$n\!=\!0.6$\ero}} % \put(961,1240){\sx{5}{\rot{64}$n\!=\!0.4$\ero}} % \put(1084,1240){\sx{5}{\rot{54}$n\!=\!0.2$\ero}} % \put(1172,1152){\sx{5.5}{\rot{44}$n\!=\!0$\ero}} \put(1230,1040){\sx{5}{\rot{34}$n\!=\!-0.2$\ero}} % \put(1210, 912){\sx{5}{\rot{26}$n\!=\!-0.4$\ero}} % \put(1204, 824){\sx{5}{\rot{19}$n\!=\!-0.6$\ero}} % \put(1200, 758){\sx{5}{\rot{15}$n\!=\!-0.8$\ero}} % \put(1210, 707){\sx{5}{\rot{11}$n\!=\!-1$\ero}} % \put(1234, 558){\sx{5}{\rot{4}$n\!=\!-2$\ero}} % \put(1234, 492){\sx{5}{\rot{2}$n\!=\!-3$\ero}} % %\put(560, 1032){\sx{6.4}{\rot{83}$y\!=\!b^{b^x}$\ero}} % \put(694, 1032){\sx{6.6}{\rot{72}$y\!=\!b^x$\ero}} % \put(870,610){\sx{6}{\rot{16}$y\!=\!\log_b(x)$\ero}} % \put(825,510){\sx{6}{\rot{6}$y\!=\!\log_b^{~2}(x)$\ero}} % \put(600,200){\sx{11}{$b\!=\!\sqrt{2}$}} \put(872,852){\sx{6}{\rot{44}$y\!=\!x$\ero}} \put(1180,374){\sx{5}{\rot{0.1}$n\!\rightarrow\!-\infty$\ero}}

\end{picture} \end{document}