File:Exm05mapT200.jpg

Complex map of the $-0.5$th iteration of exponent,

$u+\mathrm i v= \exp^{-0.5}(x+\mathrm i y)=\ln^{0.5}(x+\mathrm i y)= \mathrm{tet}(-0.5+\mathrm{ate}(x+\mathrm i y))$

For the evaluation, the non-integer iterate of exponential is expressed through tetration tet and arctetration ate. The complex double implementations FSEXP and FSLOG are used in the C++ code below.

C++ generator of curves
// Files ado.cin, conto.cin, fsexp.cin and fslog.cin should be loaded in the working directory in order to compile the code below.

using namespace std; typedef complex z_type; 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=403,N1=N+1; DB X[M1],Y[N1]; DB *g, *f, *w; // w is working array. g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB))); f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB))); w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB))); char v[M1*N1]; // v is working array FILE *o;o=fopen("exm05map.eps","w"); ado(o,802,402); fprintf(o,"401 1 translate\n 100 100 scale\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); DO(m,M1) X[m]=-4+.02*(m-.5); DO(n,N1) { y=0.+.01*(n-.5); if(y>Im(Zo)) break; Y[n]=y; } Y[n] =Im(Zo)-.00001; Y[n+1]=Im(Zo)+.00001; for(m=n+2;m.019) c=FSEXP(-.5+FSLOG(z)); p=Re(c); q=Im(c);// if(p>-19 && p<19 && ( x<2. || fabs(q)>1.e-12 && fabs(p)>1.e-12) ) { g[m*N1+n]=p;f[m*N1+n]=q;} }} fprintf(o,"1 setlinejoin 1 setlinecap\n");  p=2.;q=1; conto(o,g,w,v,X,Y,M,N, Re(Zo),-p,p);fprintf(o,".03 W 0 1 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N, Im(Zo),-p,p);fprintf(o,".03 W 0 1 0 RGB S\n"); for(m=-8;m<8;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".007 W 0 .6 0 RGB S\n"); for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".007 W .9 0 0 RGB S\n"); for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".007 W 0 0 .9 RGB S\n"); for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-q,q);fprintf(o,".02 W .8 0 0 RGB S\n"); for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-q,q);fprintf(o,".02 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".02 W .5 0 .5 RGB S\n"); for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-q,q);fprintf(o,".02 W 0 0 0 RGB S\n"); fprintf(o,"0 setlinejoin 0 setlinecap\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 "conto.cin"
 * 5) include "fsexp.cin"
 * 6) include "fslog.cin"

M(Re(Zo),Im(Zo))L(-4,Im(Zo)) fprintf(o,"1 1 1 RGB .022 W S\n"); DO(n,40){M(Re(Zo)-.2*n,Im(Zo))L(Re(Zo)-.2*(n+.4),Im(Zo)) } fprintf(o,"0 0 0 RGB .032 W S\n");

DO(n,17) {M(-4+.2*n, 0) L(-4+.2*(n+.4),0) } fprintf(o,"0 0 0 RGB .032 W S\n");

fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o); free(f); free(g); free(w); system("epstopdf exm05map.eps"); system(   "open exm05map.pdf"); //for macintosh getchar; system("killall Preview"); // For macintosh }

Latex generator of labels
% \documentclass[12pt]{article} \usepackage{geometry} \paperwidth 824pt \paperheight 426pt \usepackage{graphics} \usepackage{rotating} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \textwidth 810pt \topmargin -105pt \oddsidemargin -74pt \pagestyle{empty} \parindent 0pt \newcommand \sx {\scalebox} \begin{document} \begin{picture}(820,420) \put(20,20){\includegraphics{exm05map}} \put(4,412){\sx{2.2}{$y$}} \put(4,313){\sx{2.2}{$3$}} \put(4,213){\sx{2.2}{$2$}} \put(4,113){\sx{2.2}{$1$}} \put(4,16){\sx{2.2}{$0$}} \put(2,0){\sx{2.2}{$-4$}} \put(100,0){\sx{2.2}{$-3$}} \put(200,0){\sx{2.2}{$-2$}} \put(300,0){\sx{2.2}{$-1$}} \put(420,0){\sx{2.2}{$0$}} \put(520,0){\sx{2.2}{$1$}} \put(620,0){\sx{2.2}{$2$}} \put(720,0){\sx{2.2}{$3$}} \put(811,0){\sx{2.2}{$x$}} %\put(140,329){\sx{2.4}{$v\!=\!0$}} % \put(118,295){\sx{2.5}{\rot{-8} $u\!=\!\Re(L)$ \ero}} \put(120,235){\sx{2.5}{\rot{-4} $u\!=\!0$ \ero}} \put(120,198){\sx{2.5}{\rot{-2} $u\!=\!-0.2$ \ero}} \put(120,166){\sx{2.5}{\rot{-1} $u\!=\!-0.4$ \ero}} % \put(580,230){\sx{2.5}{\rot{ 29} $v\!=\!\Im(L)$ \ero}} \put(600,212){\sx{2.5}{\rot{ 25} $v\!=\!1.2$ \ero}} \put(616,182){\sx{2.5}{\rot{ 22} $v\!=\!1$ \ero}} % \put(86,330){\sx{2.6}{\rot{87} $v\!=\!4$ \ero}} \put(282,319){\sx{2.5}{\rot{ 69} $v\!=\!3$ \ero}} \put(520,341){\sx{2.5}{\rot{ 44} $v\!=\!2$ \ero}} \put(650,49){\sx{2.5}{\rot{ 3} $v\!=\!0.2$ \ero}} \put(650,14){\sx{2.5}{\rot{-.02} $v\!=\!0$ \ero}} % \put(28,90){\sx{2.4}{\rot{28} $u\!=\!-0.4$ \ero}} \put(140,115){\sx{2.5}{\rot{-51} $v\!=\!3$ \ero}} % \put(263,46){\sx{2.5}{\rot{28} $u\!=\!-1$ \ero}} \put(447,98){\sx{2.5}{\rot{-72} $u\!=\!0$ \ero}} \put(574,100){\sx{2.5}{\rot{-84} $u\!=\!1$ \ero}} \put(756,100){\sx{2.5}{\rot{-87} $u\!=\!2$ \ero}} % \put(53,150){\sx{2.6}{\bf cut}} \put(44,14){\sx{2.6}{\bf cut}} \end{picture} \end{document} %