File:QQFacMT.jpg

Something strange with names..

Complex map of function $H(z)=h(h(z))$, which is second iterate of the half iterate of factorial.

The half iterate is evaluatef through functions SuFac and AuFac:

$\displaystyle h(z)=\mathrm{Factorial}^{0.5}(z)=\mathrm{SuFac}\left( \frac{1}{2}+\mathrm{AuFac}(z)\right) $

The map is show with lines of constant $u$ and lines of constant $v$;

$u+\mathrm i v=H(x+\mathrm i u)$

In the right hand side of the map, it coincides with map of Factorial. The image represents figure 8.10 from the Russian version of the Book Суперфункции

C++ generator of map
Files fac.cin facp.cin afacc.cin superfactorial.cin abelfac.cin conto.cin should be loaded in order to compile the code below

//

// using namespace std; typedef std::complex z_type; //#include "sinc.cin"
 * 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 "fac.cin"
 * 1) include "facp.cin"
 * 2) include "afacc.cin"
 * 3) include "superfactorial.cin"
 * 4) include "abelfac.cin"
 * 5) 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=401,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("QQFacMap.eps","w");ado(o,402,402); fprintf(o,"201 201 translate\n 20 20 scale\n"); DO(m,M1)X[m]=-8.04+.04*(m+.5); // DO(m,M1){t=-1.+.022*m; X[m]=.2+t-1.11*exp(-1.9*t);} DO(n,N1)Y[n]=-8.04+.04*(n+.5); // DO(n,N1){t=-8.04+.04*(n+.5); t*=.97; Y[n]=t-.25*sin(0.6127874523307*t);} for(m=-8;m<9;m++){if(m==0){M(m,-8.5)L(m,8.5)} else{M(m,-8)L(m,8)}} for(n=-8;n<9;n++){    M(  -8,n)L(8,n)} fprintf(o,".008 W 0 0 0 RGB 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); //    c=afacc(z); //    c=fac(z); c=abelfac(z);  c=superfac(.5+c); c=abelfac(c);  c=superfac(.5+c); //    p=abs(c-d)/(abs(c)+abs(d));  p=-log(p)/log(10.)-1.; p=Re(c);q=Im(c); if(p>-60 && p<60 && //      (fabs(y)>.034 ||x>-.9 ||fabs(x-int(x))>1.e-3) &&           q>-60 && q<60 && fabs(q)> 1.e-16        ) {g[m*N1+n]=p;f[m*N1+n]=q;} }} //fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=1.8;q=.7; fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=1.4;q=.8; for(m=-4;m<4;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 .5 0 RGB S\n"); for(m=0;m<4;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 .8 0 0 RGB S\n"); for(m=0;m<4;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 .8 RGB S\n"); for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".04 W .8 0 0 RGB S\n"); for(m=1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-9,9); fprintf(o,".04 W .5 0 .5 RGB S\n"); for(m=-8;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n"); m=0;    conto(o,g,w,v,X,Y,M,N, (0.+m),-9,9); fprintf(o,".04 W 0 0 0 RGB S\n"); for(m=1;m<9;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n"); //#include"plofu.cin" // x=0.8856031944; // conto(o,g,w,v,X,Y,M,N,0.8856031944,-p,p); fprintf(o,".004 W .2 .2 0 RGB S\n"); M(1,0)L(-8.1,0) fprintf(o,"0 setlinejoin 0 setlinecap .05 W  1 1 1 RGB S\n"); DO(m,23){ M(1-.4*(m+.2),0)L(1-.4*(m+.4),0);} fprintf(o,".09 W 1 .4 0 RGB S\n"); DO(m,23){ M(1-.4*(m+.7),0)L(1-.4*(m+.9),0);} fprintf(o,".09 W 0 .4 1 RGB S\n"); //M(x,0)L(-8.1,0) fprintf(o,"[.19 .21]0 setdash .05 W 0 0 0 RGB S\n"); // May it be, that, some printers do not interpret well the dashing ? fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf QQFacMap.eps"); system(   "open QQFacMap.pdf");        //for LINUX //    getchar; system("killall Preview");//for mac } //

Latex generator of labels
%

\documentclass[12pt,a4paper,oneside]{book} %\newcommand \EN[1]   	% make the English version \newcommand \EN[1] –     	% suppress the English version \newcommand \RU[1]   	% make the Russian version (in this document not supported) %\newcommand \RU[1] –       	% suppress the Russian version %The Japanese version is not yet supported. While \JP is used to suppress several lines at once. \newcommand \JP[1] –    	 % ореsuppress some text %\usepackage[space]{cite }% If exist. \usepackage[utf8]{inputenc} \usepackage[T2A]{fontenc} \usepackage[russian]{babel} \usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx}

\usepackage{rotating} \usepackage{hyperref} \usepackage{wrapfig} \usepackage{geometry} %\renewcommand\baselinestretch{.94} %\renewcommand\baselinestretch{.94} %\renewcommand\baselinestretch{1.15} %\renewcommand\baselinestretch{1.2} %\renewcommand\baselinestretch{1.2} \paperwidth150mm %\paperheight 257mm %\paperheight 216mm \paperheight 147mm %\textwidth 175mm \textwidth 150mm %\oddsidemargin 5mm \oddsidemargin -72pt %\topmargin -17mm \topmargin -103pt \textheight 257mm \pagestyle{empty} \large

\usepackage{color} \definecolor{rose}{rgb}{1,.9,1} \renewcommand\thesection{\arabic{section}} \renewcommand\figurename{Рис.} %\renewcommand\refname{References} %\renewcommand\figurename{Fig.} \newcommand \be {\begin{eqnarray}} \newcommand \ee {\end{eqnarray} } \newcommand \sx {\scalebox} \newcommand \rme 	 %%makes the base of natural logarithms Roman font %\newcommand \rme 	%%makes the base of natural logarithms Italics font; choose one of these \newcommand \rmi 	 %%imaginary unity is always roman font \newcommand \ds {\displaystyle} \newcommand \bN {\mathbb{N}} \newcommand \bC {\mathbb{C}} \newcommand \bR {\mathbb{R}} \newcommand \cO {\mathcal{O}} \newcommand \cF {\mathcal{F}} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \nS {\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!} \newcommand \pS {{~}~{~}} \newcommand \fac {\mathrm{Factorial}} \newcommand {\rf}[1] {(\ref{#1})} \newcommand{\iL}[1] {~\label{#1}\pS \rm[#1]\nS}	%make the labels visible %\newcommand{\iL}[1] {\label{#1}}			%make the labels invisible; choose one of these options \newcommand \eL[1] {\iL{#1}\ee} \newcommand \ing \includegraphics \newcommand \tet {\mathrm{tet}} \parindent 0pt \begin{document} \hskip -28pt %{\sx{.72}{\begin{picture}(362,330) {\sx{1.25}{\begin{picture}(362,363) \put(0,0){\ing{QQFacMap}} \put(30,357){\sx{1.3}{$y$}} \put(30,317){\sx{1.3}{$6$}} \put(30,277){\sx{1.3}{$4$}} \put(30,237){\sx{1.3}{$2$}} \put(29,196){\sx{1.3}{$0$}} \put(20,156){\sx{1.3}{$-2$}} \put(20,116){\sx{1.3}{$-4$}} \put(20,76){\sx{1.3}{$-6$}} \put(20,36){\sx{1.2}{$-8$}} \put(70,30){\sx{1.1}{$-6$}} \put(111,30){\sx{1.1}{$-4$}} \put(151,30){\sx{1.1}{$-2$}} \put(198.6,30){\sx{1.1}{$0$}} \put(238.6,30){\sx{1.1}{$2$}} \put(278.6,30){\sx{1.1}{$4$}} \put(318.6,30){\sx{1.1}{$6$}} \put(355,30){\sx{1.1}{$x$}} \put(70,198){\sx{1.3}{\bf cut}} %central %\put(182,263){\rot{42}\sx{1.6}{\bf cut}\ero} \put(189,260){\rot{42}\sx{1.6}{\bf cut}\ero} \put(193,209){\rot{-24}\sx{1.6}{\bf cut}\ero} \put(221,290){\rot{-35}\sx{1.6}{$u\!=\!0$}\ero} \put(212,277){\rot{-40}\sx{1.6}{$v\!=\!0$}\ero} \put(199,266){\rot{-47}\sx{1.6}{$u\!=\!0$}\ero} \put(299,196.6){\sx{1.6}{$v\!=\!0$}} \end{picture}}} \end{document}

%JP{ \color{red} \begin{picture}(425,729.88) \put(0,0){\rule{1pt}{729.88pt}} \put(425,0){\rule{1pt}{729.88pt}} \put(0,0){\rule{426pt}{1pt}} \put(0,728.88){\rule{426pt}{1pt}} \end{picture} %\vskip -740pt \vskip -729pt \color{black} %} \end{document}

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