File:QQFacMT.jpg

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

The half iterate is evaluated through functions SuFac and AuFac:

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

These functions are described in the Moscow University Physics Bulletin, 2010 ^[1]

The map is shown 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, the map coincides with complex map of Factorial.

The image represents figure 8.10 from the Russian version «Суперфункции» of the Book about Superfunctions, 2014 ^[2]

This map is used as Fig.8.11 at page 102 of the English version «Superfunctions», 2020 ^[3][4]
in order to show the range of validity of relation

\[ h(h(z))=z! \]

In particular this relation holds in the range

\[ \Re(z)\!>\!1\ ,\ |\Im(z)|\!<\!4 \]

C++ generator of map

Files abelfac.cin ado.cin, afacc.cin, conto.cin, fac.cin, facp.cin, superfac.cin, should be loaded in order to compile the code below.

 #include <math.h>
 #include <stdio.h>
 #include <stdlib.h>
 #define DB double
 #define DO(x,y) for(x=0;x<y;x++)
// using namespace std;
 #include <complex>
 typedef std::complex<double> z_type;
 #define Re(x) x.real()
 #define Im(x) x.imag()
 #define I z_type(0.,1.)
 #include "fac.cin"
 //#include "sinc.cin"
 #include "facp.cin"
 #include "afacc.cin"
 #include "superfactorial.cin"
 #include "abelfac.cin"
 #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] {{#1}}   	% make the English version
\newcommand \EN[1] {{}}    	% suppress the English version
\newcommand \RU[1] {{#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 {{\rm e}}	 %%makes the base of natural logarithms Roman font 
%\newcommand \rme {{e}}	%%makes the base of natural logarithms Italics font; choose one of these
\newcommand \rmi {{\rm i}}	 %%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}
%

Warning

The code is extracted from the Book "as is". It can be significantly shortened producing the same picture.

References

Keywords

«Abelfactorial», «Abelfunction», «AuFac», «Factorial», «Iterate», «Regular iteration», «SuFac», «Superfactorial», «Superfunction», «Superfunctions», «Square root of factorial»,

«Корень из факториала», «Суперфункции»,