File:E1eiterT.jpg

Iterate of exponential to the Henryk base $\eta=\exp(1/\mathrm e)$;

id ess, the iterates of the transfer function $T(z)=\exp(z/\mathrm e)$.

The iterates are expressed through the growing superexponential $F_3$ whih is specific solution of the transfer equation

$F_3(z\!+\!1)=T( F_3(z))$

and the inverse function $G_3=F_3^{-1}$.

The graphic shows $y=T^n(x)=F_3(n+G_3(x))$

versus real $x$ for various values $n$. Thick curves correspond to the integer values of $n$

C++ generator of curves
// Files e1etf.cin, e1egf.cin, e1eti.cin, e1egi.cin should be loaded to the working directory in order to compile the code below: // //using namespace std; typedef std::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 "e1etf.cin"
 * 5) include "e1egf.cin"
 * 6) include "e1eti.cin"
 * 7) include "e1egi.cin"

void ado(FILE *O, int X, int Y) {      fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%'); fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y); fprintf(O,"/M {moveto} bind def\n"); fprintf(O,"/L {lineto} bind def\n"); fprintf(O,"/S {stroke} bind def\n"); fprintf(O,"/s {show newpath} bind def\n"); fprintf(O,"/C {closepath} bind def\n"); fprintf(O,"/F {fill} bind def\n"); fprintf(O,"/o {.1 0 360 arc C S} bind def\n"); fprintf(O,"/times-Roman findfont 20 scalefont setfont\n"); fprintf(O,"/W {setlinewidth} bind def\n"); fprintf(O,"/RGB {setrgbcolor} bind def\n");} //#include "ado.cin"


 * 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);
 * 3) define o(x,y) fprintf(o,"%6.3f %6.3f o\n",0.+x,0.+y);

int main{ int j,k,m,n; DB p,q,t1,t3,u,v,w,x,y; z_type z,c,d; FILE *o;o=fopen("e1eiter.eps","w");ado(o,1020,1020); fprintf(o,"10 10 translate\n 100 100 scale\n"); for(m=0;m<11;m++){M(m,0)L(m,10)} for(n=0;n<11;n++){M(0,n)L(10,n)}       fprintf(o,".01 W 0 0 0 RGB S\n"); //M(-3,0)L(20.2,0) M(0, -2)L(0,10.2)    fprintf(o,".03  W 0 0 0 RGB S\n"); M(0,M_E)L(M_E,M_E)L(M_E,0)                fprintf(o,".006  W 0 0 0 RGB S\n"); fprintf(o,"1 setlinejoin 1 setlinecap\n"); DO(m,101){x=.1*m; y=exp(x/M_E); if(m==0)M(x,y)else L(x,y) if(y>10.) break; } DO(m,101){x=.1*m; y=exp(exp(x/M_E)/M_E); if(m==0)M(x,y)else L(x,y) if(y>10.) break; } DO(m,101){x=.1*m; y=x; DO(n,3)y=exp(y/M_E); if(m==0)M(x,y)else L(x,y) if(y>10.) break; } DO(m,101){x=.1*m; y=x; DO(n,4)y=exp(y/M_E); if(m==0)M(x,y)else L(x,y) if(y>10.) break; } DO(m,101){x=.1*m; y=x; DO(n,5)y=exp(y/M_E); if(m==0)M(x,y)else L(x,y) if(y>10.) break; } M(0,2.71)L(M_E,M_E)L(2.72,10) fprintf(o,".05 W 0 .5 1 RGB S\n"); DO(m,200){x=1.02+.1*m; y=log(x)*M_E; if(m==0)M(x,y)else L(x,y) if(y>10.) break; } DO(m,200){x=1.4+.1*m; y=log(log(x)*M_E)*M_E; if(m==0)M(x,y)else L(x,y) if(y>10.) break; } DO(m,200){x=1.6+.1*m; y=x; DO(n,3) y=log(y)*M_E; if(m==0)M(x,y)else L(x,y) if(y>10.) break; } DO(m,200){x=1.82+.1*m; y=x; DO(n,4) y=log(y)*M_E; if(m==0)M(x,y)else L(x,y) if(y>10.) break; } DO(m,200){x=1.98+.1*m; y=x; DO(n,5) y=log(y)*M_E; if(m==0)M(x,y)else L(x,y) if(y>10.) break; } M(2.7,0)L(M_E,M_E)L(10,2.72) fprintf(o,".05 W 1 .5 0 RGB S\n"); M(0,0)L(10,10) fprintf(o,".05 W 1 0 1 RGB S\n"); for(n=-20; n< 21; n++){ DB q=.1*n; DO(m,200){x=M_E+.01+.1*m; y=Re( E1EGF(q+E1EGI(x)) ); if(m==0)M(x,y)else L(x,y) if(y>10.||x>10.) break; } } fprintf(o,".01 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n%cTrailer",'%'); fclose(o); system("epstopdf e1eiter.eps"); system(   "open e1eiter.pdf"); //mac getchar; system("killall Preview");// mac } //

Latex generator of labels
% \documentclass[12pt]{article} \usepackage{geometry} \usepackage{graphicx} \usepackage{rotating} \paperwidth 1044pt \paperheight 1050pt \topmargin -92pt \oddsidemargin -72pt \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}(1042,1024) \put(1,1016){\sx{4.2}{$y$}} \put(0,922){\sx{4.2}{$9$}} \put(0,822){\sx{4.2}{$8$}} \put(0,722){\sx{4.2}{$7$}} \put(0,622){\sx{4.2}{$6$}} \put(0,522){\sx{4.2}{$5$}} \put(0,422){\sx{4.2}{$4$}} \put(0,324){\sx{4.2}{$3$}} \put(0,294){\sx{4.2}{$\mathrm e$}} \put(0,220){\sx{4.2}{$2$}} \put(0,120){\sx{4.2}{$1$}} \put(0,20){\sx{4.2}{$0$}} \put(22,-8){\sx{4}{$0$}} \put(122,-8){\sx{4}{$1$}} \put(222,-8){\sx{4}{$2$}} \put(297,-7){\sx{4}{$\mathrm e$}} \put(322,-8){\sx{4}{$3$}} \put(422,-8){\sx{4}{$4$}} \put(522,-8){\sx{4}{$5$}} \put(622,-8){\sx{4}{$6$}} \put(722,-8){\sx{4}{$7$}} \put(822,-8){\sx{4}{$8$}} \put(922,-8){\sx{4}{$9$}} \put(1016,-7){\sx{4.1}{$x$}} %\put(0815,520){\sx{5.6}{\rot{78}$y\!=\!\exp(x)$\ero}} %\put(1118,678){\sx{4.5}{\rot{69}$y\!=\!\eta^x$\ero}} %\put(1076,606){\sx{4.1}{\rot{67}$y\!=\!\exp_{\eta}(x)$\ero}} \put(24,24){\ing{e1eiter}} %\put(10,10){\ing{expe1eplot}} \put(316,826){\sx{4.9}{\rot{90}$n\!\rightarrow\!+\infty$\ero}} \put(322, 46){\sx{4.9}{\rot{90}$n\!\rightarrow\!-\infty$\ero}} \put(416,886){\sx{4.6}{\rot{86}$n\!=\!5$\ero}} \put(476,886){\sx{4.6}{\rot{84}$n\!=\!3$\ero}} \put(524,886){\sx{4.6}{\rot{80}$n\!=\!2$\ero}} \put(630,886){\sx{4.6}{\rot{72}$n\!=\!1$\ero}} \put(708,886){\sx{4.4}{\rot{66}$n\!=\!0.6$\ero}} \put(758,886){\sx{4.4}{\rot{60}$n\!=\!0.4$\ero}} \put(810,872){\sx{4.4}{\rot{52}$n\!=\!0.2$\ero}} \put(894,876){\sx{4.6}{\rot{45}$n\!=\!0$\ero}} \put(891,792){\sx{4.4}{\rot{36}$n\!=\!-0.2$\ero}} \put(878,720){\sx{4.4}{\rot{29}$n\!=\!-0.4$\ero}} \put(878,600){\sx{4.6}{\rot{16}$n\!=\!-1$\ero}} \put(878,498){\sx{4.6}{\rot{ 7}$n\!=\!-2$\ero}} \put(878,448){\sx{4.6}{\rot{ 2}$n\!=\!-3$\ero}} \put(878,394){\sx{4.6}{\rot{1}$n\!=\!-5$\ero}} \put(70,292){\sx{4.9}{\rot{1}$n\!\rightarrow\!+\infty$\ero}} \put(830,292){\sx{4.9}{\rot{1}$n\!\rightarrow\!-\infty$\ero}} \end{picture}} \end{document}