File:Anka616map.jpg

Anka616map.jpg is complex map of Anka function

$\mathrm{Anka}(z)= \exp(\mathrm{ArcTania}(z))=\exp(z+\ln(z)-1)= \frac{1}{\mathrm e} \, z \exp(z)=\frac{1}{\mathrm e} \, \mathrm{zex}(z)$

where zex is elementary functions, $\mathrm{zex}(z)=z\,\exp(z)$

Anka appears as the Schroeder function for the transfer function Doya, and satisfies the Schroeder equation

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

Usage
Anka616map.jpg is prepared to be used as figure 16.3 in book Superfunctions .

C++ generator of map
//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=520,M1=M+1; int N=601,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("23.eps","w");ado(o,1220,1220); FILE *o;o=fopen("anka616ma.eps","w");ado(o,1220,1220); fprintf(o,"610 610 translate\n 100 100 scale\n"); DO(m,M1) X[m]=-6.+.02*(m-.5); DO(n,N1)Y[n]=-6.+.02*(n-.5); for(m=-6;m<7;m++){if(m==0){M(m,-6.2)L(m,6.2)} else{M(m,-6)L(m,6)}} for(n=-6;n<7;n++){    M(  -6,n)L(6,n)} fprintf(o,".008 W 0 0 0 RGB 2 setlinecap 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("%5.2f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); // z*=.1; c=exp(ArcTania(z)); //     c=Tania(log(z)); p=Re(c);q=Im(c); if(p>-91. && p<91. && q>-91. && q<91. ){ g[m*N1+n]=p;f[m*N1+n]=q;} }} fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=3;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,".008 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,".008 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,".008 W 0 0 .9 RGB S\n"); for(m=1;m<17;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<17;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,".03 W .6 0 .6 RGB S\n"); for(m=-16;m<17;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".03 W 0 0 0 RGB S\n"); /* y= M_PI; for(m=0;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} y=-M_PI; for(m=0;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} fprintf(o,".07 W 1 .5 0 RGB S\n"); y= M_PI; for(m=2;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} y=-M_PI; for(m=2;m<60;m+=4) {x=-7.95+.1*m; M(x,y) L(x+.05,y)} fprintf(o,".07 W 0 .5 1 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf anka616ma.eps"); system(   "open anka616ma.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{olgama}} \put(50,20){\ing{anka616ma}} \put(18,1210){\sx{5}{$y$}} \put(22,1116){\sx{4.5}{$5$}} \put(22,1016){\sx{4.5}{$4$}} \put(22,916){\sx{4.5}{$3$}} \put(22,816){\sx{4.5}{$2$}} \put(22, 716){\sx{4.5}{$1$}} \put(22, 616){\sx{4.5}{$0$}} \put(-16, 516){\sx{4.5}{$-1$}} \put(-16, 416){\sx{4.5}{$-2$}} \put(-16, 316){\sx{4.5}{$-3$}} \put(-16, 216){\sx{4.5}{$-4$}} \put(-16, 116){\sx{4.5}{$-5$}} \put(-16, 16){\sx{4.5}{$-6$}} \put(18, -20){\sx{4.5}{$-6$}} \put(118, -20){\sx{4.5}{$-5$}} \put(218, -20){\sx{4.5}{$-4$}} \put(318, -20){\sx{4.5}{$-3$}} \put(418, -20){\sx{4.5}{$-2$}} \put(518, -20){\sx{4.5}{$-1$}} \put(650, -20){\sx{4.5}{$0$}} \put(750, -20){\sx{4.5}{$1$}} \put(850, -20){\sx{4.5}{$2$}} \put(950, -20){\sx{4.5}{$3$}} \put(1050, -20){\sx{4.5}{$4$}} \put(1150, -20){\sx{4.5}{$5$}} \put(1234, -20){\sx{5}{$x$}} %\put(952, 1454){\sx{6}{\rot{75}$v\!=\!1$\ero}} %\put(914, 216){\sx{6}{\rot{-77}$v\!=\!-1$\ero}} \put(186,1180){\sx{5}{\rot{4}$u\!=\!0$\ero}} \put(184,1002){\sx{5}{\rot{5}$v\!=\!0$\ero}} \put(182,816){\sx{5}{\rot{3}$u\!=\!0$\ero}} \put(180, 616){\sx{5}{$v\!=\!0$}} % \put(1200, 614){\sx{6}{$v\!=\!0$}} \put(180, 422){\sx{5}{\rot{-6}$u\!=\!0$\ero}} \put(180, 232){\sx{5}{\rot{-6}$v\!=\!0$\ero}} \put(180, 56){\sx{5}{\rot{-6}$u\!=\!0$\ero}} % \put(564, 640){\sx{5}{\rot{59}$v\!=\!0$\ero}} \put(652, 630){\sx{5}{\rot{38}$u\!=\!0$\ero}} \put(752, 610){\sx{5}{\rot{52}$u\!=\!1$\ero}} % % \put(1066, 899){\sx{5}{\rot{2}$v\!=\!16$\ero}} \put(1066, 858){\sx{5}{\rot{3}$u\!=\!-16$\ero}} % \put(1066, 776){\sx{5}{\rot{1}$u\!=\!-16$\ero}} \put(1066, 726){\sx{5}{\rot{2}$u\!=\!16$\ero}} % \put(1066, 648){\sx{5}{\rot{-1}$v\!=\!16$\ero}}% \put(1066, 590){\sx{5}{\rot{1}$v\!=\!-16$\ero}}% % \put(1066, 518){\sx{5}{\rot{-1}$u\!=\!16$\ero}} \put(1066, 470){\sx{5}{\rot{0}$u\!=\!-16$\ero}} % \put(1060, 390){\sx{5}{\rot{-3}$v\!=\!-16$\ero}} \put(1060, 338){\sx{5}{\rot{-1}$v\!=\!16$\ero}} % \put(1040, 256){\sx{5}{\rot{-4}$u\!=\!-16$\ero}} \put(1040, 200){\sx{5}{\rot{-2}$u\!=\!16$\ero}} % \put(1030, 106){\sx{5}{\rot{-5}$v\!=\!16$\ero}} \put(1030, 60){\sx{5}{\rot{-3}$v\!=\!-16$\ero}}

\end{picture}} \end{document}