File:Aupower2map.jpg

Complex map of the Abelfunction $G$ of the power transfer function (quadratic function) $T(z)\!=\!z^2$ is shown with

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

$G(z)=\log_2(\ln(z))=\ln^2(z)/\ln(2)$

is solution of the Abel equation

$G(T(z))=G(z)+1$

Abelfunction $G$ is inverse of the superpower function

$F(z)=\exp(2^z)$

which is solution of the transfer equation

$F(z\!+\!1)=T(F(z))$

for the quadratic transfer function $T$ at base $2$.

C++ generator of map
Files ado.cin and conto.cin should be loaded in order to compile the code below. #include  typedef std::complex z_type; int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; int M=502,M1=M+1; int N=502,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("aupower2ma.eps","w");ado(o,1020,1020); fprintf(o,"510 510 translate\n 100 100 scale\n"); DO(m,M1) X[m]=-5.+.02*(m-.5); for(n=0;n<250;n++) Y[n]=-5.+.02*(n); Y[250]=-.006; Y[251]= .006; for(n=252;n
 * 2) include 
 * 3) define DB double
 * 4) define DO(x,y) for(x=0;x-99. && p<99.      &&     q>-99. && q<99.      ) {g[m*N1+n]=p; f[m*N1+n]=q; }                     }} fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1;q=.5; for(m=-11;m<11;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q, q); fprintf(o,".013 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,".012 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,".012 W 0 0 .9 RGB S\n"); for(m=1;m<11;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".028 W .9 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,".028 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".028 W .6 0 .6 RGB S\n"); for(m=-10;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".024 W 0 0 0 RGB S\n"); m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".024 W 0 0 0 RGB S\n"); for(m=1;m<11;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".024 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf aupower2ma.eps"); system(   "open aupower2ma.pdf"); getchar; system("killall Preview");//for mac }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{graphicx} \usepackage{geometry} \usepackage{rotating} \paperwidth 1044px \paperheight 1036px \topmargin -98px \oddsidemargin -90px \textwidth 2000px \textheight 2000px \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \begin{document} \begin{picture}(1030,1026) \normalsize \put(30,20){\ing{"aupower2ma"}} \put(16,1016){\sx{3}{$y$}} \put(16,918){\sx{3}{$4$}} \put(16,818){\sx{3}{$3$}} \put(16,718){\sx{3}{$2$}} \put(16,618){\sx{3}{$1$}} \put(16,518){\sx{3}{$0$}} \put(-10,418){\sx{3}{$-1$}} \put(-10,318){\sx{3}{$-2$}} \put(-10,218){\sx{3}{$-3$}} \put(-10,118){\sx{3}{$-4$}} \put(-10,18){\sx{3}{$-5$}} \put(10,-1){\sx{3}{$-5$}} \put(110,-1){\sx{3}{$-4$}} \put(210,-1){\sx{3}{$-3$}} \put(310,-1){\sx{3}{$-2$}} \put(410,-1){\sx{3}{$-1$}} \put(534,-1){\sx{3}{$0$}} \put(634,-1){\sx{3}{$1$}} \put(734,-1){\sx{3}{$2$}} \put(834,-1){\sx{3}{$3$}} \put(934,-1){\sx{3}{$4$}} \put(1028,-1){\sx{3.1}{$x$}} \put(150,522){\rot{0}{ \sx{3}{\bf cut}} \ero} %\put(150,68){\rot{0.}{ \sx{3}{$v\!=\!0$}} \ero} %\put(790,522){\rot{0}{ \sx{3}{$v\!=\!0$}} \ero} \put(246,422){\rot{34}{ \sx{3}{$u\!=\!1.6$}} \ero} \put(332,344){\rot{54}{ \sx{3}{$u\!=\!1.4$}} \ero}%% \put(295,900){\rot{84}{ \sx{3}{$u\!=\!1.4$}} \ero}% \put(430,882){\rot{63}{ \sx{3}{$u\!=\!1.2$}} \ero} \put(526,845){\rot{50}{ \sx{3}{$u\!=\!1$}} \ero} \put(490,662){\rot{-5}{ \sx{3}{$v\!=\!2$}} \ero}%% \put(734,742){\rot{90}{ \sx{3}{$v\!=\!1$}} \ero} \put(790,740){\rot{75}{ \sx{3}{$v\!=\!0.8$}} \ero} \put(846,710){\rot{56}{ \sx{3}{$v\!=\!0.6$}} \ero} \put(900,664){\rot{39}{ \sx{3}{$v\!=\!0.4$}} \ero} \put(930,600){\rot{17}{ \sx{3}{$v\!=\!0.2$}} \ero} \put(950,523){\rot{0}{ \sx{3}{$v\!=\!0$}} \ero} \put(910,453){\rot{-20}{ \sx{3}{$v\!=\!-0.2$}} \ero} \put(733,244){\rot{90}{ \sx{3}{$v\!=\!-1$}} \ero} \put(490,388){\rot{6}{ \sx{3}{$v\!=\!-2$}} \ero} % \put(746,432){\rot{65}{ \sx{3}{$u\!=\!-0.2$}} \ero} \put(778,420){\rot{57}{ \sx{3}{$u\!=\!0$}} \ero} \put(800,390){\rot{51}{ \sx{3}{$u\!=\!0.2$}} \ero} \put(846,358){\rot{50}{ \sx{3}{$u\!=\!0.4$}} \ero} \put(902,308){\rot{47}{ \sx{3}{$u\!=\!0.6$}} \ero} \put(965,227){\rot{41}{ \sx{3}{$u\!=\!0.8$}} \ero} \end{picture} \end{document}