File:Sqrt2figf45eT.png

Complex map of the real-holomorphic superfunction $F$ of the exponential to base $b\!=\!\sqrt{2}$ built up at its fixed point $L\!=\!4$ with condition $F(0)\!=\!5$.

$u\!+\!\mathrm i v=F(x\!+\!\mathrm i y)$

The image is almost Figure 2 (left at the bottom) of publication in Mathematics of Computation ^[1]; the notations are a little bit simplified.

reterences

C++ generator of curves

#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 complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#include "conto.cin"

z_type f45E(z_type z){int n; z_type e,s;
DB coeu[21]={1.,
0.44858743119526122890, .19037224679780675668, 
0.77829576536968278770e-1, 0.30935860305707997953e-1, 
0.12022125769065893274e-1, 0.45849888965617461424e-2,
0.17207423310577291102e-2, 0.63681090387985537364e-3,
0.23276960030302567773e-3, 0.84145511838119915857e-4,
0.30115646493706434120e-4, 0.10680745813035087964e-4,
0.37565713615564248453e-5, 0.13111367785052622794e-5,
0.45437916254218158081e-6, 0.15642984632975371803e-6,
0.53523276400816416929e-7, 0.18207786280204973113e-7,  
0.61604764947389226744e-8, 0.2e-8};
	e=exp(.32663425997828098238*(z-1.11520724513161));	
 	s=coeu[20]; for(n=19;n>=0;n--) { s*=e; s+=coeu[n]; } 
//	s=coeu[19]; for(n=18;n>=0;n--) { s*=e; s+=coeu[n]; } 
	return 4.+s*e;}
z_type F45E(z_type z){ DB b=sqrt(2.);
			if(Re(z)<-1.) return f45E(z);
 			return exp(F45E(z-1.)*log(b));
		}

// #include"sqrt2f45E.cin"
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
int M=501,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("sqrt2figf45e.eps","w");  ado(o,202,204);
fprintf(o,"101 101 translate\n 10 10 scale\n");
DO(m,M1) X[m]=-10.+.04*(m-.5);
//DO(n,N1) Y[n]=-10.+.04*(n-.5);
 DO(n,N1) Y[n]=2.79*sinh(2.*(n-200.5)/200.);
for(m=-10;m<11;m++) {M(m,-10)L(m,10)}
for(n=-10;n<11;n++) {M(  -10,n)L(10,n)} 
fprintf(o,"1 setlinejoin 2 setlinecap\n");
fprintf(o," .006 W 0 0 0 RGB S\n");
// z_type tm,tp,F[M1*N1];
DO(m,M1)DO(n,N1){	g[m*N1+n]=9999;
			f[m*N1+n]=9999;}
DO(m,M1){x=X[m]; 
DO(n,N1){y=Y[n]; z=z_type(x,y);	
		c=F45E(z);
		p=Re(c); 
		q=Im(c);
		if(p>-25. && p<25. && q>-25. && q<25.
		&& fabs(p)>1.e-14
		&& fabs(q)>1.e-14
		) { g[m*N1+n]=p; f[m*N1+n]=q;}
	}}
p=2.5; q=.5;
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,".01 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,".01 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,".01 W 0 0 .9 RGB S\n");
for(m= 1;m<16;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".03 W .8 0 0 RGB S\n");
for(m= 1;m<16;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".03 W 0 0 .8 RGB S\n");
               conto(o,f,w,v,X,Y,M,N, (0.  ),-2*p,2*p); fprintf(o,".03 W .5 0 .5 RGB S\n");
for(m=-31;m<16;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n");
// #include "plofu.cin"
// p=1.e-15;
//for(n=-10;n<11;n++) {q=p*n; z=z_type(q,0.); printf("%19.15f %19.16f\n",q,  Re(uq2e(z))  ); }
//M(-10,0)L(-2,0)fprintf(o,".04 W 1 0 1 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
 system("epstopdf sqrt2figf45e.eps"); 
system(    "open sqrt2figf45e.pdf"); //for macintosh
}

Latex generator of labels

% %<br> % Gerenator of sqrt2figf45eT.png %<br> % Copyleft 2011 by Dmitrii Kouznetsov %<br> \documentclass[12pt]{article} %<br> \usepackage{geometry} %<br> \usepackage{graphicx} %<br> \usepackage{rotating} %<br> \paperwidth 1050pt %<br> \paperheight 1040pt %<br> \topmargin -98pt %<br> \oddsidemargin -99pt %<br> \textwidth 1100pt %<br> \textheight 1100pt %<br> \pagestyle {empty} %<br> \newcommand \sx {\scalebox} %<br> \newcommand \rot {\begin{rotate}} %<br> \newcommand \ero {\end{rotate}} %<br> \newcommand \ing {\includegraphics} %<br> \begin{document} %<br> \sx{5}{ \begin{picture}(208,205) %<br> % \put(6,5){\ing{arctaniacontour}} %<br> \put(6,5){\ing{sqrt2figf45e}} %<br> \put(2,203.4){\sx{.7}{$y$}} %<br> \put(2,184){\sx{.6}{$8$}} %<br> \put(2,164){\sx{.6}{$6$}} %<br> \put(2,144){\sx{.6}{$4$}} %<br> \put(2,124){\sx{.6}{$2$}} %<br> %\put(23,100){\sx{.8}{$u\!=\!0$}} %<br> \put(2,104){\sx{.6}{$0$}} %<br> % \put(20, 84){\sx{.8}{$v\!=\!0$}} %<br> \put(-3,84){\sx{.6}{$-2$}} %<br> \put(-3,64){\sx{.6}{$-4$}} %<br> \put(-3,44){\sx{.6}{$-6$}} %<br> \put(-3,24){\sx{.6}{$-8$}} %<br> \put(-2,00){\sx{.6}{$-10$}} %<br> \put( 22,0){\sx{.6}{$-8$}} %<br> \put( 42,0){\sx{.6}{$-6$}} %<br> \put( 62,0){\sx{.6}{$-4$}} %<br> \put( 82,0){\sx{.6}{$-2$}} %<br> \put(106,0){\sx{.6}{$0$}} %<br> \put(126,0){\sx{.6}{$2$}} %<br> \put(146,0){\sx{.6}{$4$}} %<br> \put(166,0){\sx{.6}{$6$}} %<br> \put(186,0){\sx{.6}{$8$}} %<br> \put(203,0){\sx{.7}{$x$}} %<br> \put(20,199.4){\sx{.99}{$v\!=\!0$}} %<br> \put(20,151){\sx{.99}{$u\!=\!4$}} %<br> \put(20,103){\sx{.99}{$v\!=\!0$}} %<br> \put(20,56){\sx{.99}{$u\!=\!4$}} %<br> \put(20,07){\sx{.99}{$v\!=\!0$}} %<br> \put(072,161){\rot{46}\sx{.8}{$v\!=\!0.2$}\ero}%<br> \put(090,150){\rot{52}\sx{.8}{$v\!=\!0.4$}\ero}%<br> \put(067,114){\rot{46}\sx{.8}{$u\!=\!4.2$}\ero}%<br> \put(084,107){\rot{52}\sx{.8}{$u\!=\!4.4$}\ero}%<br> % 0 <br> \put(071,73){\rot{44}\sx{.8}{$v\!=\!-0.2$}\ero}%<br> \put(072.4, 11){\rot{66}\sx{.8}{$u\!=\!3.8$}\ero}%<br> \put(098, 11){\rot{74}\sx{.8}{$u\!=\!3.6$}\ero}%<br> \put(113.8, 11){\rot{77}\sx{.8}{$u\!=\!3.4$}\ero}%<br> \put(127, 11){\rot{80}\sx{.8}{$u\!=\!3.2$}\ero}%<br> \put(140.2, 11){\rot{88}\sx{.99}{$u\!=\!3$}\ero}%<br> \put(140.8,177){\rot{90}\sx{.99}{$u\!=\!3$}\ero}%<br> \put(178,150){\rot{9}\sx{.99}{$u\!=\!2$}\ero}%<br> \put(128,154){\rot{0.}\sx{.99}{$v\!=\!1$}\ero}%<br> \put(128,136){\rot{0.}\sx{.99}{$v\!=\!2$}\ero}%<br> \put(125,054){\rot{0.}\sx{.99}{$v\!=\!-1$}\ero}%<br> \put(178,057){\rot{-10}\sx{.99}{$u\!=\!2$}\ero}%<br> \end{picture} %<br> } %<br> \end{document}