File:Olga6map.jpg

Complex map of the Olga function,

$u+\mathrm i v= \mathrm{Olga}(x\!+\!\mathrm i y)$

Olga is scaling function for the Doya function; it is solution of the scaling equation

$\mathrm{Doya}(\mathrm{Olga}(z))=\mathrm e\, \mathrm{Olga}(z)$

Olga can be expressed through the Tania function:

$\mathrm{Olga}(z)=\mathrm{Tania}(\ln(z))$

This representation is used for the C++ implementation below.

Usage
This image is prepared to be used as figure 6.4 in the English version of book Superfunctions

C++ generator of map
Files ado.cin and conto.cin should be loaded in order to compile the C++ code below:

//using namespace std; typedef std::complex z_type; z_type ArcTania(z_type z) {return z + log(z) - 1. ;} z_type ArcTaniap(z_type z) {return 1. + 1./z ;} z_type TaniaTay(z_type z) { int n; z_type s; s=1.+z*(.5+z*(1./16.+z*(-1./192.+z*(-1./3072.+z*(1.3/6144.+z*(-4.7/147456. //+z*(7.3/4128768.) //some reserve term )))))); DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; } z_type TaniaNega(z_type z){int n;z_type s=exp(z-exp(z)+1.); DO(n,4) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; } z_type TaniaBig(z_type z){int n;z_type s=z; s=z-log(s)+1.; DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; } z_type TaniaS(z_type z){int n; z_type s,t=z+z_type(2.,-M_PI);t*=2./9.; t=I*sqrt(t); s=-1.+t*(3.+t*(-3.+t*(.75+t*(.3+t*(.9/16.+t*(-.3/7.+t*(-12.51/224. //+t*(-.9/28.) ))))))); DO(n,3) s+=(z-ArcTania(s))/ArcTaniap(s); return s ; } z_type Tania(z_type z){ z_type t; if( fabs(Im(z))< M_PI && Re(z)<-2.51) return TaniaNega(z); if( abs(z)>7. || Re(z)>3.8 ) return TaniaBig(z); if( Im(z) > .7 ) return TaniaS(z); if( Im(z) < -.7) return conj(TaniaS(conj(z))); return TaniaTay(z); }
 * 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"

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=603,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("taniacontour.eps","w");ado(o,1620,1620); //FILE *o;o=fopen("13.eps","w");ado(o,1620,1620); FILE *o;o=fopen("olga6ma.eps","w");ado(o,1220,1220); fprintf(o,"610 610 translate\n 100 100 scale\n"); DO(m,M1) X[m]=-6.+.03*(m); DO(n,300)Y[n]=-6.+.02*n; Y[300]=-.012; Y[301]=-.006; Y[302]= .006; Y[303]= .012; for(n=304;n-99. && p<99. && q>-99. && q<99. ){ g[m*N1+n]=p;f[m*N1+n]=q;} }} fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=.6;q=.5; 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,".009 W 0 .6 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,".009 W .9 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,".009 W 0 0 .9 RGB S\n"); for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".03 W .9 0 0 RGB S\n"); for(m=1;m<10;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".03 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".01 W .6 0 .6 RGB S\n"); for(m=-9;m<10;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".03 W 0 0 0 RGB S\n");

// M(-10.*exp(-2.),0)L(-8,0) M(-exp(-2.),0)L(-8,0) fprintf(o,"0 setlinecap .01 W 1 1 1 RGB S\n");

y= 0; for(m=1;m<80;m+=4) {x=-.1353352832366127-.1*m; M(x,y) L(x-.08,y)} fprintf(o,".05 W 1 .5 0 RGB S\n"); y= 0; for(m=3;m<80;m+=4) {x=-.1353352832366127-.1*m; M(x,y) L(x-.08,y)} fprintf(o,".05 W 0 .5 1 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf olga6ma.eps"); system(   "open olga6ma.pdf"); getchar; system("killall Preview"); }

Latex generator of labels
\documentclass[12pt]{article} \paperwidth 1268px \paperheight 1260px \textwidth 1794px \textheight 1700px \topmargin -98px \oddsidemargin -70px \usepackage{graphics} \usepackage{rotating} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \rmi {\mathrm{i}} \begin{document} \parindent 0pt \sx{1}{\begin{picture}(1252,1230) %\put(40,20){\ing{suzexD1map}} %\put(40,20){\ing{ZexD6map}} \put(50,20){\ing{olga6ma}} \put(14,1204){\sx{6}{$y$}} \put(18,1116){\sx{5}{$5$}} \put(18,1016){\sx{5}{$4$}} \put(18, 916){\sx{5}{$3$}} \put(18, 816){\sx{5}{$2$}} \put(18, 716){\sx{5}{$1$}} \put(18, 616){\sx{5}{$0$}} \put(-22, 516){\sx{5}{$-1$}} \put(-22, 416){\sx{5}{$-2$}} \put(-22, 316){\sx{5}{$-3$}} \put(-22, 216){\sx{5}{$-4$}} \put(-22, 116){\sx{5}{$-5$}} \put(-22, 016){\sx{5}{$-6$}} \put( 18, -20){\sx{5}{$-6$}} \put(118, -20){\sx{5}{$-5$}} \put(218, -20){\sx{5}{$-4$}} \put(318, -20){\sx{5}{$-3$}} \put(418, -20){\sx{5}{$-2$}} \put(518, -20){\sx{5}{$-1$}} \put(649, -20){\sx{5}{$0$}} \put(750, -20){\sx{5}{$1$}} \put(850, -20){\sx{5}{$2$}} \put(950, -20){\sx{5}{$3$}} \put(1050, -20){\sx{5}{$4$}} \put(1150, -20){\sx{5}{$5$}} \put(1234, -20){\sx{6}{$x$}} \put(170, 616){\sx{6}{\bf cut}} \put(1060, 614){\sx{6}{$v\!=\!0$}} \put(710, 1086){\sx{6}{\rot{75}$v\!=\!1$\ero}} \put(660, 222){\sx{6}{\rot{-77}$v\!=\!-1$\ero}} \put(86,790){\sx{6}{\rot{-29}$v\!=\!2$\ero}} \put(100, 438){\sx{6}{\rot{28}$v\!=\!-2$\ero}} % \put(78, 1050){\sx{6}{\rot{37}$u\!=\!2$\ero}} \put(460, 668){\sx{6}{\rot{38}$u\!=\!1$\ero}}

\end{picture}} \end{document}