File:AuPow2Plot.jpg

Explicit plof of two abelpower functions:

$y\!=$AuPow$_2(x)\!=\! \log_2\!\big(\ln(x)\big)~$, blue curve

and

$y\!=$AdPow$_2(x)\!=\! \log_2\!\big(\ln(1/x)\big)~$, red curve

C++ generator of curves
File ado.cin should be loaded in order to compile the code below. //#using namespace std; //#include //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) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "ado.cin"

DB B=2.; DB F(DB z) { return exp( exp( log(B)*z));} DB G(DB z) { return log( log(z) )/log(B);} DB H(DB z) { return log( log(1./z))/log(B);}

DB T(DB z) { return exp(B*log(z));} DB U(DB z) { return exp(log(z)/B);}

int main{ int m,n; double x,y,t; FILE *o; o=fopen("aupow2plo.eps","w"); ado(o,720,740); fprintf(o,"10 510 translate 100 100 scale\n");
 * 1) define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);

fprintf(o,"2 setlinecap 1 setlinejoin .03 W 0 0 1 RGB S\n");

for(m=0;m<8;m++) {M(m,-5)L(m,2)} for(n=-5;n<3;n++) {M(0,n)L(7,n)} fprintf(o,".01 W 0 0 0 RGB S\n");

DO(m,98){x=.01+.01*m; y=H(x); if(m==0)M(x,y) else L(x,y);} fprintf(o,".04 W 1 0 0 RGB S\n");

DO(m,601){x=1.03+.01*m; y=G(x); if(m==0)M(x,y) else L(x,y);} fprintf(o,".04 W 0 0 1 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf aupow2plo.eps"); system(   "open aupow2plo.pdf"); getchar; system("killall Preview"); }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{geometry} \usepackage{graphicx} \usepackage{rotating} \paperwidth 760pt \paperheight 708pt \topmargin -94pt \oddsidemargin -81pt \textwidth 1100pt \textheight 1100pt \pagestyle {empty} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \parindent 0pt \pagestyle{empty} \begin{document} \begin{picture}(742,702) %\put(51,1){\ing{01plo}} \put(51,1){\ing{aupow2plo}} \put(24,696){\sx{4}{$y$}} \put(24,598){\sx{4}{$1$}} \put(24,498){\sx{4}{$0$}} \put(-4,398){\sx{4}{$-1$}} \put(-4,298){\sx{4}{$-2$}}

\put(-4,198){\sx{4}{$-3$}} \put(-4,098){\sx{4}{$-4$}} \put(150,476){\sx{4}{$1$}} \put(250,476){\sx{4}{$2$}} \put(351,476){\sx{4}{$3$}} \put(451,476){\sx{4}{$4$}} \put(552,476){\sx{4}{$5$}} \put(652,476){\sx{4}{$6$}} \put(738,476){\sx{4}{$x$}} \put(366,546){\sx{3.7}{\rot{13}$y\!=\!\mathrm{AuPow}_2(x)$\ero}} %\put(242,66){\sx{3.8}{\rot{82}$y\!=\!\mathrm{AuPow}_2(x)$\ero}} \put(80,666){\sx{3.7}{\rot{-77}$y\!=\!\mathrm{AdPow}_2(x)$\ero}} \end{picture} \end{document}