File:ZexIteT.jpg

Explicit plot of iterations of function Zex

$ y=\mathrm{zex}^n(x)$

for various number $n$ of iterates. Here, $\mathrm{zex}(z)\!=\!z\exp(z)$. The non-integer iterates are implemented through the superfunciton of zex and its abelfunction, named SuZex and AuZex, as follows:

$ \mathrm{zex}^n(x)=\mathrm{SuZex}\Big(n+\mathrm{AuZex}(x)\Big)$

C++ generator of curves
// Files ado.cin, Tania.cin, LambertW.cin, SuZex.cin, AuZex.cin should be loaded to the working directory in order to compile the 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.)


 * 1) include "Tania.cin" // need for LambertW
 * 2) include "LambertW.cin" // need for AuZex
 * 3) include "SuZex.cin"
 * 4) include "AuZex.cin"
 * 5) include "ado.cin"
 * 6) define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
 * 7) define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);

main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; FILE *o;o=fopen("ZexIte.eps","w");  ado(o,1204,1204); fprintf(o,"2 2 translate\n 100 100 scale\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); for(n=0;n<13;n++) {M(0,n)L(12,n)} for(m=0;m<13;m++) {M(m,0)L(m,12)} M(M_E,0)L(M_E,1) M(0,M_E)L(1,M_E) fprintf(o,".01 W S\n"); DO(m,700){x=.01 +.02*m; y=Re(LambertW(LambertW(x)));if(m==0) M(x,y) else L(x,y) if(x>12.03||y>12.03) break;} fprintf(o,".033 W 1 0 1 RGB S\n"); DO(m,700){x=.01 +.02*m; y=Re(LambertW(x));if(m==0) M(x,y) else L(x,y) if(x>12.03||y>12.03) break;} fprintf(o,".04 W 1 0 1 RGB S\n"); M(0,0) L(12.03,12.03) fprintf(o,".03 W 0 1 0 RGB S\n"); DO(m,700){x=.01 +.02*m; y=Re(zex(x));    if(m==0) M(x,y) else L(x,y) if(x>12.03||y>12.03) break;} fprintf(o,".04 W 0 1 0 RGB S\n"); DO(m,700){x=.01 +.02*m; y=Re(zex(zex(x)));    if(m==0) M(x,y) else L(x,y) if(x>12.03||y>12.03) break;} fprintf(o,".033 W 0 1 0 RGB S\n");

for(n=-10;n<11;n++){ DO(m,700){x=.01 +.02*m; y=Re(auzex(x)); y=Re(suzex(.1*n+y)); if(m==0) M(x,y) else L(x,y) if(x>12.03||y>12.03) break;} fprintf(o,".023 W 0 0 0 RGB S\n"); } fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o); system("epstopdf ZexIte.eps"); system(   "open ZexIte.pdf"); //for macintosh getchar; system("killall Preview"); // For macintosh }

Latex generator of labels
% % % % file ZexIte.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 1208pt % \paperheight 1208pt % \topmargin -103pt % \oddsidemargin -73pt % \textwidth 1404pt % \textheight 1404pt % \pagestyle {empty} % \newcommand \sx {\scalebox} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \ing {\includegraphics} % \parindent 0pt% \pagestyle{empty} % \begin{document} % \begin{picture}(1202,1202) % %\put(10,10){\ing{IterPowPlot}} % %\put(10,10){\ing{IterEq2plot}} % \put(0,0){\ing{ZexIte}} % \put(11,1184){\sx{4.4}{$y$}} % \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(1180,6){\sx{4.4}{$x$}} % % \put(116,1058){\sx{5}{\rot{88}$n\!=\!2$\ero}} % \put(182,1058){\sx{5}{\rot{88}$n\!=\!1$\ero}} % \put(141,706){\sx{4.5}{\rot{86}$y\!=\!\mathrm{zex}(x)$\ero}} % % \put(512,1030){\sx{5}{\rot{72}$n\!=\!0.3$\ero}} % \put(629,1032){\sx{5}{\rot{64}$n\!=\!0.2$\ero}} % \put(804,1037){\sx{5}{\rot{56}$n\!=\!0.1$\ero}} % % \put(1072,1052){\sx{5}{\rot{44}$n\!=\!0$\ero}} % \put(772,752){\sx{5}{\rot{44}$y\!=\!x$\ero}} % % \put(1028,762){\sx{5}{\rot{32}$n\!=\!-0.1$\ero}} % \put(1014,590){\sx{5}{\rot{23}$n\!=\!-0.2$\ero}} % \put(1006,470){\sx{5}{\rot{17}$n\!=\!-0.3$\ero}} % % \put(510,108){\sx{4.3}{\rot{3}$y\!=\!\mathrm{LambertW}(x)$\ero}} % \put(1010,142){\sx{5}{\rot{2}$n\!=\!-1$\ero}} % \put(1010,83){\sx{5}{\rot{1}$n\!=\!-2$\ero}} % % \end{picture} % \end{document} % %