File:Kneserplot.png

Explicit plot of the Keneser function (red curve),

\( \varphi=\mathrm{Kneser}=\exp^{1/2} \)

The Keneser function is solution \( \varphi \) of equation

\( \varphi(\varphi(z)) = \exp(z) \)

Description
The Kneser function is implemented through the natural tetration tet and the arctetration ate:

\( \varphi(z)=\mathrm{tet}(1/2+\mathrm{ate}(z)) \)

Few additional lines are drown:

The short segment along line \( x\!=\!\mathrm{tet}(-1.5) \) and Horisontal asymptotic \( y\!=\!\mathrm{tet}(-1.5) \)

Bisection of the First quadrant \( y\!=\!x \)

Exponential \( y\!=\!\exp(x) \)

C++ generator of map
// files ado.cin, conto.cin, fsexp.cin, fslog.cin should be loaded using namespace std; typedef complex z_type; int main{ int j,k,m,n; DB x,y; z_type c,z; FILE *o; o=fopen("kneserplo.eps","w"); ado(o,1220,920); fprintf(o,"810 110 translate\n 100 100 scale\n"); for(m=-8;m<5;m++) {M(m,-1)L(m,8)} for(n=-1;n<9;n++) {M( -8,n)L(4,n)} fprintf(o,".006 W 0 0 0 RGB S\n");
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x8.) break;};

fprintf(o,"0 setlinecap 0 0 1 RGB .08 W S\n");

for(m=0;m<121;m++){ x=.1*(m-80); z=x; c=FSLOG(z); c=FSEXP(.5+c); y=Re(c); if(m==0) M(x,y) else L(x,y) if(y>8.) break;} fprintf(o,"2 setlinecap 1 setlinejoin 1 0 0 RGB .08 W S\n");

z=-1.5; y=Re(FSEXP(z));

printf("y= %9.6lf\n",y);

M(-8,y)L(0,y) M(y,y)L(y,0) fprintf(o,"0 setlinecap 0 0 0 RGB .007 W S\n");

fprintf(o,"showpage\n"); fprintf(o,"%cTrailer\n",'%'); fclose(o); system("epstopdf kneserplo.eps"); system(   "open kneserplo.pdf"); //for macintosh getchar; system("killall Preview"); // For macintosh }

Latex generator of curves
% File kneserplo.pdf should be generated with the cone above

\documentclass[12pt]{article} \usepackage{geometry} \paperwidth 1300pt \paperheight 974pt \textheight 1800pt \textwidth 1800pt \topmargin -88pt \oddsidemargin -72pt \usepackage{graphics} \newcommand \sx {\scalebox} \newcommand \ing {\includegraphics} \usepackage{rotating} \newcommand \rot {\begin{turn}} \newcommand \ero {\end{turn}} \pagestyle{empty} \parindent 0pt \begin{document} \huge \begin{picture}(1020,920) \put(80,20){\ing{kneserplo}} \put(40,906){\sx{3}{$y$}} \put(40,710){\sx{3}{$6$}} \put(40,510){\sx{3}{$4$}} \put(40,310){\sx{3}{$2$}} \put(40,110){\sx{3}{$0$}} \put(24,-28){\sx{3}{$-8$}} \put(224,-28){\sx{3}{$-6$}} \put(424,-28){\sx{3}{$-4$}} \put(624,-28){\sx{3}{$-2$}} \put(880,-28){\sx{3}{$0$}} \put(1080,-28){\sx{3}{$2$}} \put(1260,-26){\sx{3.2}{$x$}} \put(960,550){\rot{81}\sx{3.1}{$y\!=\!\exp(x)$}\ero} \put(1086,418){\rot{69}\sx{3.1}{$y\!=\!\mathrm{Kneser}(x)$}\ero} \put(1150,356){\rot{45}\sx{3.1}{$y\!=\!x$}\ero} \put(894,50){\rot{0}\sx{3}{$y\!=\!\mathrm{tet}(-1.5)$}\ero} \end{picture} \end{document}