File:E1eghalfm3.jpg

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Complex map of upper half iterate of exponential to base $\eta=\exp(1/\mathrm e)$ [1]:

$u\!+\!\mathrm i v=\exp_{\eta,u}^{1/2}(x\!+\!\mathrm i y)$

where $\exp_{\eta,\mathrm u}^{1/2}(z)=$SuExp$_{\eta}\big(1/2+$SuExp$_{\eta}(z)\big)$

C++ generator of map

// files ado.cin, conto.cin, e1egf.cin, e1egi.cin should be loaded in order to compile the code below

//


#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
//using namespace std;
#include <complex>
typedef std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "e1egf.cin"
#include "e1egi.cin"
#include "conto.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
 int M=401,M1=M+1;
 int N=301,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("e1eghalf.eps","w");ado(o,402,302);
fprintf(o,"101 151 translate\n 10 10 scale\n");
DO(m,M1) X[m]=-10.+.1*(m-.5);
DO(n,N1) Y[n]=-15.+.1*(n-.5);
for(m=-10;m<31;m++){ //if(m==0){ M(m,-10.2)L(m,10.2)} else
                                { M(m,-15)L(m,15) }}
for(n=-15;n<16;n++){ M(-10,n)L(30,n)}
fprintf(o,".006 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
DO(m,M1){x=X[m]; printf("run at x=%6.3f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
        c=E1EGF(.5+E1EGI(z));
// c=E1EGF(.5+E1EGI(c));
// d=z;
// p=abs(c-d)/abs(c+d); p=-log(p)/log(10.);
        p=Re(c); q=Im(c);
// if(p>-85 && p<85) g[m*N1+n]=p;
        if(p>-101.3 && p<101.3 && fabs(p)> 1.e-11 && //fabs(p-1.)>1.e-9 &&
          q >-101.3 && q<101.3 && fabs(q)> 1.e-11) { g[m*N1+n]=p; f[m*N1+n]=q; }
        }}

fprintf(o,"1 setlinejoin 1 setlinecap\n");
//p=1.;q=.5;
//#include"plofu.cin"
p=4.;q=.5;

for(m= 1;m<11;m++) conto(o,f,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".04 W .8 0 0 RGB S\n");
for(m= 1;m<11;m++) conto(o,f,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 .8 RGB S\n");
                   conto(o,f,w,v,X,Y,M,N,(0. ),-p,p);fprintf(o,".04 W .5 0 .5 RGB S\n");
for(m=-10;m<11;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 0 RGB S\n");

m=15;
conto(o,f,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".04 W .8 0 0 RGB S\n");
conto(o,f,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 .8 RGB S\n");
      conto(o,g,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".04 W 0 0 0 RGB S\n");
      conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 0 RGB S\n");

for(m=20; m<101;m+=10){
 conto(o,f,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".04 W .8 0 0 RGB S\n");
 conto(o,f,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 .8 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".04 W 0 0 0 RGB S\n");
 conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 0 RGB S\n");
}
/*
m=30; conto(o,f,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".04 W .8 0 0 RGB S\n");
      conto(o,f,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 .8 RGB S\n");
      conto(o,g,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".04 W 0 0 0 RGB S\n");
      conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 0 RGB S\n");
*/
DO(m,M1){
DO(n,N1){ if( fabs(g[m*N1+n])>5. || fabs(f[m*N1+n])>5. )
                { g[m*N1+n]=999.;f[m*N1+n]=999.;}
        }}

for(m=-5;m<6;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".01 W 0 .6 0 RGB S\n");
for(m=0;m<5;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);
                                                fprintf(o,".01 W .9 0 0 RGB S\n");
for(m=0;m<5;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);
                                                fprintf(o,".01 W 0 0 .9 RGB S\n");
/*
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");
*/
fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
        system("epstopdf e1eghalf.eps");
        system( "open e1eghalf.pdf"); //mac
// system( "xpdf e1eghalf.pdf"); // linux
getchar(); system("killall Preview");// mac
}

Latex generator of labels


\documentclass[12pt]{article}
\usepackage{graphicx}
\usepackage{rotating}
\usepackage{geometry}
\paperwidth 420px
\paperheight 322px
\topmargin -107pt
\oddsidemargin -80pt
\textheight 600px
\pagestyle{empty}
\begin{document}
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}

\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}

\newcommand \figaxeh {
\put( -5,306){\sx{1}{$y$}}
\put( -10,289){\sx{1}{$14$}}
\put( -10,269){\sx{1}{$12$}}
\put( -10,249){\sx{1}{$10$}}
\put( -6,229){\sx{1}{$8$}}
\put( -6,209){\sx{1}{$6$}}
\put( -6,189){\sx{1}{$4$}}
\put( -6,169){\sx{1}{$2$}}
\put( -6, 149){\sx{1}{$0$}}
\put(-14,129){\sx{1}{$-2$}}
\put(-14,109){\sx{1}{$-4$}}
\put(-14, 89){\sx{1}{$-6$}}
\put(-14, 69){\sx{1}{$-8$}}
\put(-18, 49){\sx{1}{$-10$}}
\put(-18, 29){\sx{1}{$-12$}}
\put(-18, 9){\sx{1}{$-14$}}
%\put(-13, 79){\sx{1}{$-6$}}
\put(400,-6.5){\sx{1}{$x$}}
\put(379,-7){\sx{.9}{$28$}}
\put(359,-7){\sx{.9}{$26$}}
\put(339,-7){\sx{.9}{$24$}}
\put(319,-7){\sx{.9}{$22$}}
\put(299,-7){\sx{.9}{$20$}}
\put(279,-7){\sx{.9}{$18$}}
\put(259,-7){\sx{.9}{$16$}}
\put(239,-7){\sx{.9}{$14$}}
\put(219,-7){\sx{.9}{$12$}}
\put(199,-7){\sx{.9}{$10$}}
\put(181,-7){\sx{.9}{$8$}}
\put(161,-7){\sx{.9}{$6$}}
\put(141,-7){\sx{.9}{$4$}}
\put(121,-7){\sx{.9}{$2$}}
\put(101,-7){\sx{.9}{$0$}}
\put( 75,-7){\sx{.9}{$-2$}}
\put( 55,-7){\sx{.9}{$-4$}}
\put( 35,-7){\sx{.9}{$-6$}}
\put( 15,-7){\sx{.9}{$-8$}}
}
\begin{picture}(262,312)
\put(2,2){\ing{e1eghalf}}
\figaxeh
%\put( 21,182){\sx{1.3}{\rot{53} \textcolor{white}{\rule{26pt}{9pt}}\ero}}
\put( 18,182){\sx{1.3}{\rot{49} $v\!=\!-1$\ero}}
%\put( 61,173){\sx{1.3}{\rot{85} \textcolor{white}{\rule{23pt}{9pt}}\ero}}
\put( 61,175){\sx{1.3}{\rot{82} $v\!=\!0$\ero}}
\put(16,150.2){\sx{1.4}{\bf cut - - - - - - -}}
%\put(257,148.4){\sx{1.3}{$q\!=\!0$}}
%\put(307,148.4){\sx{1.7}{$\exp_{b,3}^{[0.5]}(z)\!\rightarrow\!\infty$}}
\put(316,297){\sx{1.1}{\rot{16}$u\!=\!10$\ero}}
\put(323,288){\sx{1.1}{\rot{20}$u\!=\!20$\ero}}
% \put(338,274){\sx{1.1}{\rot{40}$u\!=\!40$\ero}}
\put(343,266){\sx{1.1}{\rot{45}$u\!=\!50$\ero}}
\put(358,254){\sx{1.1}{\rot{51}$u\!=\!70$\ero}}
\put(379,242){\sx{1.1}{\rot{54}$u\!=\!100$\ero}}

\put(373,236){\sx{1.1}{\rot{-18}$v\!=\!100$\ero}}
\put(357,198){\sx{1.1}{\rot{-9}$v\!=\!50$\ero}}
\put(350,170){\sx{1.1}{\rot{-2}$v\!=\!20$\ero}}
\put(349,160){\sx{1.1}{\rot{-1}$v\!=\!10$\ero}}
\put(349,150){\sx{1.1}{$v\!=\!0$}}
\put(349,141){\sx{1.1}{\rot{1}$v\!=\!-10$\ero}}
\put(350,131){\sx{1.1}{\rot{2}$v\!=\!-20$\ero}}
\put(357,104){\sx{1.1}{\rot{9}$v\!=\!-50$\ero}}
%\put(142,225){\sx{.9}{\rot{-52} \textcolor{white}{\rule{22pt}{9pt}} \ero }}
\put(166,249){\sx{1.}{\rot{-60} $v\!=\!15$\ero}}
\put(144,229){\sx{1.}{\rot{-54} $v\!=\!9$\ero}}
\put(130,218){\sx{1.}{\rot{-53} $v\!=\!6$\ero}}
%\put( 99,175){\sx{1.2}{ \rot{45} \textcolor{white}{\rule{19pt}{10pt}} \ero }}
\put( 99,174){\sx{1.2}{\rot{45} $u\!=\!0$\ero}}
%\put(117,165){\sx{1.2}{\rot{57} \textcolor{white}{\rule{20pt}{9pt}} \ero }}
\put(113,166){\sx{1.2}{\rot{57} $u\!=\!1$\ero}}
%\put(136,155){\sx{1.2}{\rot{72} \textcolor{white}{\rule{21pt}{9pt}} \ero }}
\put(136,156){\sx{1.2}{\rot{72} $u\!=\!3$\ero}}
\end{picture}
\end{document}

References

  1. ↑ http://mizugadro.mydns.jp/PAPERS/2012e1eMcom2590.pdf H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics of Computation, v.81 (2012), p. 2207-2227.

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