File:Apow2ma4.jpg

Two complex maps of the real-holomorphic abelpower functions,

AuPow$_2(z)\!=\!\log_2(\ln(z))~$ and $~$ AdPow$_2(z)\!=\!\log_2(\ln(1/z))$

Left: $u\!+\!\mathrm i v=\mathrm{AuPow}_2(x\!+\!\mathrm i y)$

Right: $u\!+\!\mathrm i v=\mathrm{AdPow}_2(x\!+\!\mathrm i y)$

Description
The two real–holomorphic solutions $G$ of the Abel equation

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

for $T(z)\!=\!z^2$ are shown.

Lines of the constant real part are the same for both maps.

The imaginary parts differ for a constant, but this constant has jumps at the cutlines. All these cutlines are at the real axis.

Both function have cut for negative values of argument. Each of them has also one additional cut:

AuPow has cut segment between zero and unity.

AdPow has cut from unity to infinity.

Many other abelfunctions for the same transfer function can be obtained adding to the function some periodic function with period unity. Such functions show fast (at least exponential) growth in the imaginary direction; the abelfunctions shown seem to be simplest possible.

C++ generator of left map
// ado.cin and conto.cin should be loaded for compilation of the codes below: 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=402,M1=M+1; int N=402,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("aupow2ma4.eps","w");ado(o,420,420); fprintf(o,"210 210 translate\n 100 100 scale\n"); DO(m,M1) X[m]=-2.+.01*(m-.5); for(n=0;n<250;n++) Y[n]=-2.+.01*(n); Y[200]=-.006; Y[201]= .006; for(n=202;n
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) 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,".01 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,".01 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,".01 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,".02 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,".02 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".02 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,".02 W 0 0 0 RGB S\n"); m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 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,".02 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf aupow2ma4.eps"); system( "open aupow2ma4.pdf"); getchar; system("killall Preview");//for mac }

C++ generator of right map
ado.cin and conto.cin should be loaded 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=402,M1=M+1; int N=402,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("adpow2ma4.eps","w");ado(o,420,420); fprintf(o,"210 210 translate\n 100 100 scale\n"); DO(m,M1) X[m]=-2.+.01*(m-.5); for(n=0;n<250;n++) Y[n]=-2.+.01*(n); Y[200]=-.006; Y[201]= .006; for(n=202;n
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) 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,".01 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,".01 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,".01 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,".02 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,".02 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".02 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,".02 W 0 0 0 RGB S\n"); m=0; conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".02 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,".02 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf adpow2ma4.eps"); system( "open adpow2ma4.pdf"); getchar; system("killall Preview");//for mac }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{graphicx} \usepackage{geometry} \usepackage{rotating} %\paperwidth 432px \paperwidth 854px \paperheight 422px \topmargin -98px \oddsidemargin -90px \textwidth 2000px \textheight 2000px \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \axey { \put(12,412){\sx{2}{$y$}} \put(12,312){\sx{2}{$1$}} \put(12,212){\sx{2}{$0$}} \put(-3,112){\sx{2}{$-1$}} \put(-3,11){\sx{2}{$-2$}}} \newcommand \axexb { \put( 13, 1){\sx{2}{$-2$}} \put(112, 1){\sx{2}{$-1$}} \put(225, 1){\sx{2}{$0$}} \put(325, 1){\sx{2}{$1$}} \put(419, 1){\sx{2.1}{$x$}} \put(32, 252){\rot{-18}{ \sx{1.8}{$u\!=\!1.6$}} \ero} \put( 68,216){\rot{0}{ \sx{1.6}{\bf cut}} \ero} \put(32, 178){\rot{18}{ \sx{1.8}{$u\!=\!1.6$}} \ero} \put(36, 64){\rot{51}{ \sx{1.8}{$u\!=\!1.4$}} \ero} \put(109, 23){\rot{71}{ \sx{1.8}{$u\!=\!1.2$}} \ero} \put(174, 23){\rot{86}{ \sx{1.9}{$u\!=\!1$}} \ero} \put(348,104){\rot{11}{ \sx{1.8}{$u\!=\!-0.2$}} \ero} \put(354,83){\rot{0}{ \sx{1.8}{$u\!=\!0$}} \ero} \put(348,55){\rot{-5}{ \sx{1.8}{$u\!=\!0.2$}} \ero} } \begin{document} \begin{picture}(420,413) \normalsize \put(20,10){\ing{"aupow2ma4"}} \axey \axexb \put(240,351){\rot{-27}{ \sx{1.8}{$v\!=\!2$}} \ero} \put(240,266){\rot{-6}{ \sx{1.8}{$v\!=\!3$}} \ero} \put(242,229){\rot{0}{ \sx{1.8}{$v\!=\!4$}} \ero} \put(390,290){\rot{69}{ \sx{1.8}{$v\!=\!1$}} \ero} \put(380,215.3){\rot{0}{ \sx{1.78}{$v\!=\!0$}} \ero} \put(358,187){\rot{-63}{ \sx{1.78}{$v\!=\!-1$}} \ero} \put(210, 76){\rot{16}{ \sx{1.8}{$v\!=\!-2$}} \ero} \end{picture} \begin{picture}(420,413) \normalsize \put(20,10){\ing{"adpow2ma4"}} \axexb \put( 396,216){\rot{0}{ \sx{1.6}{\bf cut}} \ero} \put(241,215){\rot{0}{ \sx{1.8}{$v\!=\!0$}} \ero} \put(236,189){\rot{-2}{ \sx{1.8}{$v\!=\!1$}} \ero} \put(216,142){\rot{-5}{ \sx{1.8}{$v\!=\!2$}} \ero} \put(300, 28){\rot{51}{ \sx{1.8}{$v\!=\!3$}} \ero}

\put(383,240){\rot{35}{ \sx{1.8}{$v\!=\!-4$}} \ero} \put(384,190){\rot{-35}{ \sx{1.8}{$v\!=\!4$}} \ero} \end{picture} \end{document}