File:ExpIte4T.jpg

iterations of exp: $y\!=\!\exp^n(x)$ for various values of $n$.

For non-integer values of $n$, the evaluation is performed using tetration and ArcTetration funcitons,

$y\!=\!\exp^n(x)=\mathrm{tet}\Big(n+\mathrm{ate}(x)\Big)$

C++ generator of curves
// Files ado.cin, fsexp.cin, fslog.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.)

//#include "Tania.cin" // need for LambertW //#include "LambertW.cin" // need for AuZex //#include "SuZex.cin" //#include "AuZex.cin"
 * 1) include "fsexp.cin"
 * 2) include "fslog.cin"

// z_type tra(z_type z){ return exp(z)+z;} // z_type F(z_type z){ return log(suzex(z));} // z_type G(z_type z){ return auzex(exp(z));}


 * 1) include "ado.cin"
 * 2) define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
 * 3) 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("ExpIte4.eps","w");  ado(o,804,804); fprintf(o,"402 402 translate\n 100 100 scale\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); for(n=-4;n<5;n++) {M(-4,n)L(4,n)} for(m=-4;m<5;m++) {M(m,-4)L(m,4)} // M(M_E,0)L(M_E,1) M(0,M_E)L(1,M_E) fprintf(o,".004 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 0 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 .5 0 RGB S\n"); // M(0,0) L(12.03,12.03) fprintf(o,".04 W 0 1 0 RGB S\n"); DO(m,700){x=-4.02+.02*m; y=exp(x);         if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;} fprintf(o,".032 W 0 1 0 RGB S\n"); DO(m,700){x=-4.02+.02*m; y=exp(exp(x));    if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;} fprintf(o,".032 W 0 1 0 RGB S\n"); DO(m,700){x=-4.02+.02*m; y=exp(exp(exp(x)));if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;} fprintf(o,".032 W 0 1 0 RGB S\n"); DO(m,700){y=-4.02+.02*m; x=exp(y);         if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;} fprintf(o,".032 W 1 0 1 RGB S\n"); DO(m,700){y=-4.02+.02*m; x=exp(exp(y));    if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;} fprintf(o,".032 W 1 0 1 RGB S\n"); DO(m,700){y=-4.02+.02*m; x=exp(exp(exp(y)));if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;} fprintf(o,".032 W 1 0 1 RGB S\n"); for(n=0;n<34;n+=1) {DO(m,700){x=-4.01 +.02*m; y=Re(FSLOG(x)); y=Re(FSEXP(.1*n+y)); if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;}} for(n=-33;n<0;n+=1){t=Re(FSEXP( FSLOG(-4.)-.1*n)); DO(m,700){x=t +.02*m; y=Re(FSLOG(x)); y=Re(FSEXP(.1*n+y)); if(m==0) M(x,y) else L(x,y) if(x>4.03||y>4.03) break;}} fprintf(o,".01 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o); system("epstopdf ExpIte4.eps"); system(   "open ExpIte4.pdf"); //for macintosh getchar; system("killall Preview"); // For macintosh }

Latex Generator of labels
% % % file *.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 806pt % \paperheight 806pt % \topmargin -105pt % \oddsidemargin -73pt % \textwidth 1100pt % \textheight 1100pt % \pagestyle {empty} % \newcommand \sx {\scalebox} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \ing {\includegraphics} % \parindent 0pt% \pagestyle{empty} % \begin{document} % \begin{picture}(802,802) % %\put(10,10){\ing{PowPlo}} % %\put(0,0){\ing{TraItu3}} % \put(0,0){\ing{ExpIte4}} % \put(411,788){\sx{3}{$y$}} % \put(411,693.6){\sx{2.9}{$3$}} % \put(411,593.4){\sx{2.9}{$2$}} % \put(411,493.2){\sx{2.9}{$1$}} % \put(411,393){\sx{2.9}{$0$}} % \put(407,292.8){\sx{2.9}{$-1$}} % \put(407,192.6){\sx{2.9}{$-2$}} % \put(407,092.4){\sx{2.9}{$-3$}} % % \put(081,408){\sx{2.9}{$-3$}} % \put(181,408){\sx{2.9}{$-2$}} % \put(281,408){\sx{2.9}{$-1$}} % \put(396,408){\sx{2.9}{$0$}} % \put(497,408){\sx{2.9}{$1$}} % \put(597,408){\sx{2.9}{$2$}} % \put(697,408){\sx{2.9}{$3$}} % \put(787,408){\sx{3}{$x$}} % % \put(6,748){\sx{3}{\rot{4}$n\!=\!3.2$\ero}} % \put(6,708){\sx{3}{\rot{3}$n\!=\!3.1$\ero}} % \put(6,675){\sx{3}{\rot{3}$n\!=\!3$\ero}} % \put(5,643){\sx{3}{\rot{3}$n\!=\!2.9$\ero}} % \put(6,345){\sx{3}{\rot{1}$n\!=\!0.6$\ero}} % % \put(7,308){\sx{3}{\rot{4}$n\!=\!0.4$\ero}} % \put(7,280){\sx{3}{\rot{5}$n\!=\!0.3$\ero}} % \put(8,242){\sx{3}{\rot{9}$n\!=\!0.2$\ero}} % \put(9,185){\sx{3}{\rot{11}$n\!=\!0.1$\ero}} % % \put(50,36){\sx{3}{\rot{45}$n\!=\!0$\ero}} % % \put(202,5){\sx{3}{\rot{76}$n\!=\!-0.1$\ero}} % \put(263,5){\sx{3}{\rot{82}$n\!=\!-0.2$\ero}} % \put(299,5){\sx{3}{\rot{84}$n\!=\!-0.3$\ero}} % \put(691,5){\sx{3}{\rot{84}$n\!=\!-3$\ero}} % \put(724,5){\sx{3}{\rot{83}$n\!=\!-3.1$\ero}} % \put(764,5){\sx{3}{\rot{82}$n\!=\!-3.2$\ero}} % % \put(480,600){\sx{3.6}{\rot{69}$y\!=\!\exp(x)$\ero}} % \put(641,630){\sx{3.4}{\rot{44}$y\!=\!x$\ero}} % \put(650,484){\sx{3.6}{\rot{19}$y\!=\!\ln(x)$\ero}} %

\end{picture} % \end{document} % %