File:Exp1exp2t.jpg

Iterate of exponent (thin black lines) compared to the linar combination of $\exp$ and $\exp^2$, deawn with thick green lines.

Iterates of exponent are evaluated through tetration tet and arctetration ate:

$y=\exp^n(x)=\mathrm{tet}(n+\mathrm{ate}(x))$, thin lines

The corresponding linear combinations of the first and the second iterates of the exponent are defined with

$y=(2\!-\!n)\exp(x)+(n\!-\!1)\exp(\exp(x))$, and marked with thick green lines.

Description
This image is modification of figure 15.4 from the book Суперфункции, 2014

The English translation Superfunctions in 2015 yet is in preparation

Evaluation of the natural tetration tet and the arctetration ate is described also in the Mathematics of computation (2009) and in the Vladicavkaz mathematical journal (2010) .

C++ generator of curves
Files ado.cin, fsexp.cin, fslog.cin should be loaded into working directory in order to compile the code below //using namespace std; typedef std::complex z_type; //#include "Tania.cin" // need for LambertW //#include "LambertW.cin" // need for AuZex //#include "SuZex.cin" //#include "AuZex.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));} int 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); FILE *o;o=fopen("exp1exp2.eps","w"); ado(o,604,404); fprintf(o,"402 2 translate\n 100 100 scale\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); for(n=0;n<5;n++) {M(-4,n)L(2,n)} for(m=-4;m<3;m++) {M(m,0)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");
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x4.03) 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=10;n<21;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 exp1exp2.eps"); system(   "open exp1exp2.pdf"); //for macintosh //     getchar; system("killall Preview"); // For macintosh return 0; }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \usepackage{graphicx} \usepackage{rotating} \paperwidth 616pt \paperheight 422pt \topmargin -108pt \oddsidemargin -72pt \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}(614,414) %\put(0,0){\ing{ExpIte4}} \put(10,10){\ing{exp1exp2}} \put(1,400){\sx{1.8}{$y = \exp^n(x)$, thin lines}} \put(1,376){\sx{1.8}{and $(2\!-\!n)\exp(x)+(n\!-\!1)\exp(\exp(x))$,}} \put(270,350){\sx{1.8}{thick lines}} \put(-1,306.6){\sx{2.}{$3$}} \put(-1,206.4){\sx{2.}{$2$}} \put(-1,106){\sx{2.}{$1$}} \put(-1, 6){\sx{2.}{$0$}}

\put(094,-6){\sx{2}{$-3$}} \put(194,-6){\sx{2}{$-2$}} \put(294,-6){\sx{2}{$-1$}} \put(409,-6){\sx{2}{$0$}} \put(509,-6){\sx{2}{$1$}} \put(600,-5){\sx{2}{$x$}} % \put(20,118){\sx{1.8}{\rot{0}$n\!=\!2$\ero}} \put(20,99){\sx{1.8}{\rot{0}$n\!=\!1.9$\ero}} %\put(20,88){\sx{1.8}{\rot{0}$n\!=\!1.8$\ero}} \put(20, 79){\sx{1.8}{\rot{0}$n\!=\!1.7$\ero}} \put(20, 58){\sx{1.8}{\rot{0}$n\!=\!1.5$\ero}} \put(20, 39){\sx{1.8}{\rot{0}$n\!=\!1.3$\ero}} %\put(20, 29){\sx{1.8}{\rot{0}$n\!=\!1.2$\ero}} \put(20, 19){\sx{1.8}{\rot{0}$n\!=\!1.1$\ero}} \put(20, -1){\sx{1.8}{\rot{0}$n\!=\!1$\ero}} % \put(398,260){\sx{2}{\rot{74}$y\!=\!\exp(\exp(x))$\ero}} \put(504,220){\sx{2}{\rot{66}$y\!=\!\exp(x)$\ero}}

\end{picture} \end{document}

Remarks
This plot is generated by request by Ю.