File:B271t.png

Complex map of tetration to base $\mathrm e$, isolines of real and imaginary parts of
 * $f\!=\!\mathrm{tet}(x + \mathrm i y)$ are shown in the $x$,$y$ plane,

range: $~-4\! \le\! x \le 4~$; $~-4\! \le\! y \le 4~$.

Levels $u\!=\!\Re(f)\!=\!\mathrm {const}$ and $v\!=\!\Im(f)\!=\!\mathrm {const}$ are plotted; integer values correspond to the thick lines.

C++ generator of curves
// Files conto.cin, ado.cin, fsexp.cin, plofu.cin should be loaded to the working directory for the compilation of the code below.

using namespace std; typedef complex z_type; //#include "fslog.cin" main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; int M=401,M1=M+1; int N=401,N1=N+1; DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array. char v[M1*N1]; // v is working array FILE *o;o=fopen("b271t0.eps","w"); ado(o,87,87); fprintf(o,"46 45 translate\n 10 10 scale\n"); DO(m,M1) X[m]=-4.+.02*(m-.5); DO(n,N1) Y[n]=-4.+.02*(n-.5); for(m=-4;m<5;m++) {M(m,-4)L(m,4)} for(n=-4;n<5;n++) {M( -4,n)L(4,n)} fprintf(o,".006 W 0 0 0 RGB S\n"); fprintf(o,"/adobe-Roman findfont .6 scalefont setfont\n"); DO(m,M1)DO(n,N1){     g[m*N1+n]=9999; f[m*N1+n]=9999;} DB b=sqrt(2); DO(m,M1){x=X[m]; printf("x=%6.3f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); if(abs(z+2.)>.04) { c=FSEXP(z); //      c=FSLOG(z); p=Re(c); q=Im(c); if(p>-9999 && p<9999 && fabs(q)>1.e-12) g[m*N1+n]=p; if(q>-9999 && q<9999 && fabs(q)>1.e-12) f[m*N1+n]=q; }        }} fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=2.; q=1.1;; fprintf(o,"0 setlinejoin 0 setlinecap\n"); fprintf(o,"showpage\n"); fprintf(o,"%cTrailer\n",'%'); fclose(o); system("epstopdf b271t0.eps"); system(   "open b271t0.pdf"); //for macintosh getchar; system("killall Preview"); // For macintosh }
 * 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 "conto.cin"
 * 5) include "fsexp.cin"
 * 1) include "plofu.cin"

Latex generator of labels
\documentclass[12pt]{article} % \paperheight 680px % \paperwidth 772px % \textwidth 1294px % \textheight 1200px % \topmargin -110px % \oddsidemargin -80px % \usepackage{graphics} % \usepackage{rotating} % \newcommand \sx {\scalebox} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \ing {\includegraphics} % \newcommand \rmi {\mathrm{i}} % \begin{document} % \newcommand \zoomax { % \put(2, 82.6){\sx{.6}{$y$}} % \put(2, 63.3){\sx{.5}{$2$}} % \put(2, 43.3){\sx{.5}{$0$}} % \put(-1, 23.3){\sx{.5}{$-\!2$}} % \put(22, 0.8){\sx{.5}{$-\!2$}} % \put(45, 0.8){\sx{.5}{$0$}} % \put(65, 0.8){\sx{.5}{$2$}} % \put(84,1){\sx{.5}{$x$}} % } % \parindent 0pt % \sx{8}{\begin{picture}(86,86) \put(0,0){\ing{b271t0}} % \zoomax % \put(7,44){\sx{.4}{\bf cut}} % \put(7,81){\sx{.4}{$f\approx L\approx0.318+1.337i$}} % \put(6,6){\sx{.4}{$f\approx L^*\approx0.318-1.337i$}} % \put(87,78){\sx{.31}{$u\!=\!0.4$}} % \put(87,74){\sx{.31}{$v\!=\!1.2$}} % \put(87,68){\sx{.32}{$v\!=\!1$}} % \put(87,56){\sx{.32}{$v\!=\!0$}} % \put(87,44){\sx{.32}{$v\!=\!0$}} % \put(87,33){\sx{.31}{$v\!=\!0$}} % \put(87,21){\sx{.31}{$v\!=\!-\!1$}} % \put(86,14){\sx{.29}{$v\!=\!-\!1.2$}} % \put(87, 11){\sx{.30}{$u\!=\!0.4$}} % \put(10.2,66){\sx{.3}{\rot{2}$v\!=\!1.4$\ero}}   \put(55.2,77){\sx{.3}{\rot{2}$v\!=\!1.4$\ero}} % \put(10,22){\sx{.3}{\rot{-2}$v\!=\!-\!1.4$\ero}}  \put(55,12){\sx{.3}{\rot{-2}$v\!=\!-\!1.4$\ero}} % \put(22,64){\sx{.3}{\rot{15}$u\!=\!0.2$\ero}}  \put(41,71){\sx{.3}{\rot{16}$u\!=\!0.4$\ero}} % \put(22,25){\sx{.3}{\rot{-15}$u\!=\!0.2$\ero}} \put(41,18){\sx{.3}{\rot{-16}$u\!=\!0.4$\ero}} % \end{picture}} % \end{document}