File:Sqrt2uiimap80.jpg

Complex map of iterate number i of exponent to base $\sqrt{2}$ constructed at its upper fixed point 4:

$u\!+\!\mathrm i v= \exp_{\sqrt{2},\mathrm u}(x\!+\!\mathrm i y)$

Usage: this is figure 16.12 of the book Суперфункции (2014, In Russian) ; the English version is in preparation in 2015. (Numeration of figures in the English version may be different from that of the Russian version.)

The algorithm of the evaluation is also described in the article . (top right map)

C++ generator of the map
Files ado.cin, conto.cin, sqrt2f21e.cin, sqrt2f21l.cin, sqrt2f45e.cin, sqrt2f45l.cin should be loaded in order to compile the code below.

// using namespace std; typedef std::complex z_type;
 * 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) include "conto.cin"


 * 1) include "sqrt2f45e.cin"
 * 2) include "sqrt2f45l.cin"

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; int M=801,M1=M+1; int N=405,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("sqrt2uiima.eps","w"); ado(o,202,202); fprintf(o,"101 101 translate\n 10 10 scale\n"); DO(m,M1) X[m]=-10.+.025*(m-.5); //DO(n,N1) Y[n]=-10.+.04*(n-.5); // DO(n,200) Y[n]=sinh(3.*(n-200.5)/200.); // DO(n,200) Y[n]=-10.+.05*(n-.5); //        Y[200]=-.0001; //        Y[201]= .0001; for(n=0;n-201. && p<201. && q>-201. && q<201. //             && fabs(p)>1.e-14 //              && fabs(q)>1.e-14                ) { g[m*N1+n]=p; f[m*N1+n]=q;} }} p=10; q=1; for(m=-10;m<10;m++)for(n=2             ;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q);  fprintf(o,".014 W 0 .7 0 RGB S\n"); for(m=0;m<11;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);                 fprintf(o,".014 W 1 0 0 RGB S\n"); for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);                   fprintf(o,".014 W 0 0 1 RGB S\n"); for(m= 1;m<100;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<100;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=-100;m<101;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" M(-10,0)L(2,0)fprintf(o,"0 setlinecap .036 W 1 1 1 RGB S\n"); for(n=0;n<27;n++){ M(2-.5*(n+.2),0) L(2-.5*(n+.4),0) } fprintf(o,".06 W 1 .5 0 RGB S\n"); for(n=0;n<27;n++){ M(2-.5*(n+.7),0) L(2-.5*(n+.9),0) } fprintf(o,".06 W 0 .5 1 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); // system("epstopdf 03.eps"); system("epstopdf sqrt2uiima.eps"); system(   "open sqrt2uiima.pdf"); //for macintosh }

Latex generator of labels
File generated with code above should be loaded in order to compile the code below.

\documentclass[12pt]{article} \paperwidth 2072px \paperheight 2076px \textwidth 2394px \textheight 2300px \topmargin -97px \oddsidemargin -78px \usepackage{graphics} \usepackage{rotating} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \rmi {\mathrm{i}} \parindent 0pt \pagestyle{empty} \begin{document}\parindent 0pt \sx{10}{\begin{picture}(206,206) %\put(6,5){\ing{Esqrt2ite13Map}} %\put(6,5){\ing{03}} \put(6,5){\ing{sqrt2uiima}} \put(2,203.4){\sx{.7}{$y$}} \put(2,184){\sx{.6}{$8$}} \put(2,164){\sx{.6}{$6$}} \put(2,144){\sx{.6}{$4$}} \put(2,124){\sx{.6}{$2$}} \put(2,104){\sx{.6}{$0$}} \put(-2.2,84){\sx{.6}{$-2$}} \put(-2.2,64){\sx{.6}{$-4$}} \put(-2.2,44){\sx{.6}{$-6$}} \put(-2.2,24){\sx{.6}{$-8$}} \put(-2,-1){\sx{.7}{$-\!10$}} \put( 22,-1){\sx{.7}{$-8$}} \put( 42,-1){\sx{.7}{$-6$}} \put( 62,-1){\sx{.7}{$-4$}} \put( 82,-1){\sx{.7}{$-2$}} \put(106,-1){\sx{.7}{$0$}} \put(126,-1){\sx{.7}{$2$}} \put(146,-1){\sx{.7}{$4$}} \put(166,-1){\sx{.7}{$6$}} \put(186,-1){\sx{.7}{$8$}} \put(204,-1){\sx{.7}{$x$}} \put(024,103.5){\sx{.99}{\bf cut}} \put(118,173){\sx{.99}{\rot{-16}$v\!=\!3$\ero}} \put(146,135){\sx{.99}{\rot{-29}$v\!=\!2$\ero}} \put(148,117){\sx{.99}{\rot{-33}$v\!=\!1$\ero}} \put(149,105){\sx{.99}{\rot{-40}$v\!=\!0$\ero}} \put(149,096){\sx{.99}{\rot{-56}$v\!=\!-1$\ero}} % \put(57,140){\sx{.99}{\rot{-47}$u\!=\!1$\ero}} % \put(26,30){\sx{.99}{\rot{84}$v\!=\!100$\ero}} \put(37,57){\sx{.99}{\rot{-74}$v\!=\!0$\ero}} \put(39,83){\sx{.99}{\rot{-66}$v\!=\!-100$\ero}} % \put(51,99){\sx{.99}{\rot{-49}$u\!=\!-100$\ero}} \put(53,87){\sx{.99}{\rot{-64}$u\!=\!100$\ero}} % \put(115.6,180){\sx{.99}{\rot{74}$u\!=\!2$\ero}} \put(155,170){\sx{.99}{\rot{47}$u\!=\!3$\ero}} \put(174.4,147.6){\sx{.99}{\rot{36}$u\!=\!4$\ero}} \put(176,122){\sx{.99}{\rot{31}$u\!=\!5$\ero}} \put(176,104){\sx{.99}{\rot{26}$u\!=\!6$\ero}} \put(176,092){\sx{.99}{\rot{17}$u\!=\!7$\ero}} \put(176,082){\sx{.99}{\rot{12}$u\!=\!8$\ero}} \put(176,074){\sx{.99}{\rot{4}$u\!=\!9$\ero}} \put(176,067){\sx{.99}{\rot{-2}$u\!=\!10$\ero}} \put(176,055){\sx{.99}{\rot{-16}$u\!=\!12$\ero}} \put(176,044){\sx{.99}{\rot{-28}$u\!=\!14$\ero}} \put(176,033){\sx{.99}{\rot{-40}$u\!=\!16$\ero}} \put(169,027){\sx{.99}{\rot{-48}$u\!=\!18$\ero}} \put(159,025){\sx{.96}{\rot{-50}$u\!=\!20$\ero}} \end{picture}} \end{document}