File:Expe1eplotT.jpg

Explicit plot of exponential to base e1e (thick green curve) and that of the exponential to base sqrt2 (thin red curve)

Here, $\eta\!=\!\exp(1\mathrm e)\!\approx1.44466786$\! is the Henryk base. At this base the exponential has only one real fixed point, id est,  equation $\exp_\eta(L)\!=\!L$ has only one real solution $L\!=\!\mathrm e\!\approx\! 2.71~$.

The thick green curve is $!y\!=\!\eta^x$.

In order to show the fixed point, the thin line $y\!=\!x$ is drawn.

For comparison, the exponential to base $b\!=\! \sqrt{2}$ is plotted, that has two fixed points, $L\!=\!2$ and $L\!=\!4$.

C++ generator of curves
//
 * 1) include
 * 2) include
 * 3) include
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include "ado.cin"

DB B=sqrt(2.);

int main{ int m,n; double x,y; FILE *o; o=fopen("expe1eplot.eps","w"); ado(o,1204,804); fprintf(o,"602 2 translate 100 100 scale\n"); for(m=-6;m<7;m++) {M(m,0)L(m,8)} for(m=0;m<9;m++) {M(-6,m)L(6,m)} fprintf(o,"2 setlinecap .01 W S\n 1 setlinejoin \n"); M(M_E,0)L(M_E,M_E)L(0,M_E) fprintf(o,".007 W S\n");
 * 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);

for(m=0;m<123;m++){x=-6.1+.1*m; y=exp(log(B)*x); if(m==0)M(x,y) else L(x,y);} fprintf(o,".02 W .8 0 0 RGB S\n");

for(m=0;m<123;m++){x=-6.1+.1*m; y=exp(x/M_E); if(m==0)M(x,y) else L(x,y);} fprintf(o,".04 W 0 .6 0 RGB S\n");

M(-.1,-.1)L(6.1,6.1) fprintf(o,".016 W 0 0 0 RGB S\n\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf expe1eplot.eps"); system(   "open expe1eplot.pdf"); getchar; system("killall Preview");//for mac } //

Latex generator of labels
% \documentclass[12pt]{article} \usepackage{geometry} \usepackage{graphicx} \usepackage{rotating} \paperwidth 1212pt \paperheight 844pt \topmargin -92pt \oddsidemargin -80pt \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}(1202,802) \put(590,792){\sx{4.2}{$y$}} \put(590,698){\sx{4.2}{$7$}} \put(590,598){\sx{4.2}{$6$}} \put(590,498){\sx{4.2}{$5$}} \put(590,398){\sx{4.2}{$4$}} \put(590,298){\sx{4.2}{$3$}} \put(620,274){\sx{4.2}{$\mathrm e$}} \put(590,198){\sx{4.2}{$2$}} \put(590,098){\sx{4.2}{$1$}} \put(080,-22){\sx{4}{$-5$}} \put(180,-22){\sx{4}{$-4$}} \put(281,-22){\sx{4}{$-3$}} \put(381,-22){\sx{4}{$-2$}} \put(482,-22){\sx{4}{$-\!1$}} \put(603.6,-22){\sx{4}{$0$}} \put(703.7,-22){\sx{4}{$1$}} \put(803.8,-22){\sx{4}{$2$}} \put(877.,16){\sx{4}{$\mathrm e$}} \put(903.9,-22){\sx{4}{$3$}} \put(1004.0,-22){\sx{4}{$4$}} \put(1104.1,-22){\sx{4}{$5$}} \put(1192.2,-22){\sx{4.3}{$x$}} %\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(1100,520){\sx{4}{\rot{62}$y\!=\!\exp_{_{\!\!\sqrt{2}}}(x)$\ero}} \put(1130,550){\sx{4}{\rot{61}$y\!=\!(\sqrt{2})^x$\ero}} \put(1134,488){\sx{5}{\rot{45.1}$y\!=\!x$\ero}} \put(10,10){\ing{expe1eplot}} \end{picture}} \end{document}