File:E1eplot8.png

Explicit plot of the two superexponentials to base $\eta=\exp(1/\mathrm e)$

Description
Superexponential to base $b$ is solution $F$ of the transfer equation

$F(z+1)=\exp_b(F(z))$

For $b=\eta=\exp(1/\mathrm e)$, The solutions that have specific asymptotic behaviour are considered, they approadh the fixed point $\mathrm e\approx 2.71$ ; these solutions are described in.

Use of this image
Figure 1 of the article COMPUTATION OF THE TWO REGULAR SUPER-EXPONENTIALS TO BASE EXP(1/E) .

Figure 22.4 of the book Суперфункции

C++ generator of curves
//using namespace std; typedef std::complex z_type; 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" 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("fige1e1.eps","w");ado(o,234,124); FILE *o;o=fopen("e1eplot.eps","w");ado(o,234,124); fprintf(o,"32 22 translate\n 10 10 scale\n"); for(m=-3;m<21;m++){if(m!=0){M(m,-2)L(m,10)}} for(n= -2;n<11;n++){if(n!=0){M(-3,n)L(20,n)}}  fprintf(o,".006 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(-3,M_E)L(20.3,M_E)                           fprintf(o,".05  W 0 .8 0 RGB S\n");
 * 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"
 * 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);

fprintf(o,"1 setlinejoin 1 setlinecap\n");

DO(m,227){x=-3+.1*m; y=Re(E1EGF(x));   if(m==0)M(x,y)else L(x,y)} fprintf(o,".05 W 0 0 1 RGB S\n");

//DO(m,220){x=-1.5+.1*m; y=Re(E1ETF(x));       if(m==0)M(x,y)else L(x,y)} fprintf(o,".05 W 1 0 0 RGB S\n"); DO(m,221){x=-1.66+.1*m; y=Re(E1ETF(x)); if(m==0)M(x,y)else L(x,y)} fprintf(o,".05 W 1 0 0 RGB S\n");

DO(n,61){y=-2+.2*n; z=z_type(y,M_E*M_PI); x=Re(E1ETI(z)); printf("%9.4f %9.4f\n",x,y); if(n==0)M(x,y)else L(x,y)} fprintf(o,".05 W .7 0 .7 RGB S\n"); fprintf(o,".03 W 0 0 0 RGB\n"); t1=-2.7982482;

/* FILE *i; i=fopen("walket1.txt","r"); DO(k,30){ j=fscanf(i,"%lf%lf%lf",&u,&v,&w); printf("%2d %5.2lf %14.10lf %14.10lf\n",j,u,v,w); if(j<3) break; if(v>10) break; o(v+t1,u) } printf("\n");

fclose(i); i=fopen("walket2.txt","r"); DO(k,30){ j=fscanf(i,"%lf%lf%lf%lf",&y,&u,&v,&w);  if(j<4) break; printf("%5.2f %5.2lf %14.10lf %14.10lf\n",y,u,v,w); o(w+t1,y); }

printf("\n"); t3=20.2874; i=fopen("walket3.txt","r"); DO(k,30){ j=fscanf(i,"%lf%lf%lf",&u,&v,&w); printf("%2d %5.2lf %14.10lf %14.10lf\n",j,u,v,w); if(j<3) break; if(u>-10)      o(v+t3,u) } fclose(i);

//fprintf(o,"0 setlinecap\n"); M(-2,0)L(-8,0) fprintf(o,".08 W 1 1 1 RGB S\n"); //DO(m,16){M(-2-.4*(m),0)L(-2-.4*(m+.5),0)} fprintf(o,".09 W 0 0 0 RGB S\n"); //M(-2,0)L(-10,0) fprintf(o,".04 W 0 0 0 RGB [.1 .1] 1 setdash S\n");

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o); system("epstopdf e1eplot.eps"); system(   "open e1eplot.pdf"); //mac //     system(    "xpdf e1eplot.pdf"); // linux

//q=1.e-14; //for(n=-10;n<11;n++){z=q*n; y=Re(E1EGF(z)); //printf("E1EGF(%20.17f)=%20.17f\n",q*n,y);}

getchar; system("killall Preview");// mac }

Latex generator of labels
\documentclass[12pt]{article} %\paperwidth 472px %\paperheight 800px \paperwidth 232px \paperheight 122px \textwidth 704px \textheight 900px \topmargin -111px \oddsidemargin -73.4px \usepackage{graphics} \usepackage{rotating} \usepackage[usenames]{color} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \rmi {\mathrm{i}} \parindent 0pt \pagestyle{empty} \begin{document} \begin{picture}(240,126) %\put(0,0){\ing{fige1efre}} %\put(0,0){\ing{fige1ew13}} \put(0,0){\ing{e1eplot}} \put( 26,119){\sx{.8}{$y$}} \put( 27,100){\sx{.7}{$8$}} \put( 27,80){\sx{.7}{$6$}} \put( 27,60){\sx{.7}{$4$}} \put( 27,40){\sx{.7}{$2$}} \put( 27,20){\sx{.7}{$0$}} \put( 30,14){\sx{.7}{$0$}} \put( 50,14){\sx{.7}{$2$}} \put( 70,14){\sx{.7}{$4$}} \put( 90,14){\sx{.7}{$6$}} \put(110,14){\sx{.7}{$8$}} \put(128,14){\sx{.7}{$10$}} \put(149,14){\sx{.7}{$12$}} \put(169,14){\sx{.7}{$14$}} \put(189,14){\sx{.7}{$16$}} \put(209,14){\sx{.7}{$18$}} \put(229,14){\sx{.8}{$x$}} \put(6,120){\sx{.7}{\rot{-88}$y\!=\!\Re\!\Big(F_{1}(x\!+\!\rmi o)\Big)$\ero}} %\put(183,95){\sx{1.}{$y\!=\!F_{3}(x)$}} %\put(193,52){\sx{1.}{$y\!=\!\rme$}} %\put(193,40){\sx{1.}{$y\!=\!F_{1}(x)$}} %Dima's new latex does not like this %\put(183,95){\sx{.8}{$y\!=\!F_{3}(x)$}} %\put(136,38){\sx{.8}{$y\!=\!\mathrm{SuExp}_{\eta,3}(x)\!=\!F_{1}(x)$}} %\put(148,95){\sx{.8}{$y\!=\!\mathrm{tet}_\eta(x)\!=\!F_{3}(x)$}} \put(126,95){\sx{.8}{$y\!=\!\mathrm{SuExp}_{\eta,3}(x)\!=\!F_{3}(x)$}} \put(148,38){\sx{.8}{$y\!=\!\mathrm{tet}_\eta(x)\!=\!F_{1}(x)$}} \put(193,52){\sx{.8}{$y\!=\!\mathrm e$}} \end{picture} \end{document}