File:Factorialz.jpg

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Factorial in the complex plane.

\[ u+\mathrm iv = \mathrm{Factorial}(x\!+\!\mathrm iy) \]

The additional level \(u=0.8856031944\) corresponds to the local minimum of Factorial of positive argument.

Superfunctions

This map appears as Fig.8.3 at page 92 of book «Superfunctions»[1][2]
as a test of the compex double implementation of Factorial, but also to show its behavior in the complex plane.
In the Book, the Factorial is interpreted as a Transferfunction.
For this function, the Superfunction \(\mathrm{SuFac}\) and the Abelfunction \(\mathrm{AuFac}=\mathrm{SuFac}^{-1}\) and constructed. Then the \(n\)th iterate of Factorial ie expressed as follows: \[ \mathrm{Factorial}^n=\mathrm{SuFac}\big(n+\mathrm{AuFac}(x)\big) \] In particular, at \(n!=\!1/2\), this formula expresses the Square root of factorial [3]; since century 20, it is used as LOGO of the Physics department of the Moscow State Univeristy.

Sitizendium

Similar picture appears at http://en.citizendium.org/wiki/Image:Factorialz.jpg

C++ generator of map

// files fac.cin, ado.cin, conto.cin should be loaded 

 #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 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 "superfac.cin"
 #include "conto.cin"

 main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
   int M=403,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("fig2b.eps","w");ado(o,402,402);
 FILE *o;o=fopen("facmap.eps","w");ado(o,402,402);
 fprintf(o,"201 201 translate\n 20 20 scale\n");
 DO(m,212) X[m]=-8.+.04*(m);
    X[212]=.45;
    X[213]=.46;
    X[214]=.47;
 for(m=215;m<M1;m++) X[m]=-8.+.04*(m-3.); 
 DO(n,200)Y[n]=-8.+.04*n;
         Y[200]=-.008;
         Y[201]= .008;
 for(n=202;n<N1;n++) Y[n]=-8.+.04*(n-1.); 
 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=superfac(z);
 //	p=abs(c-d)/(abs(c)+abs(d));  p=-log(p)/log(10.)-1.;
 	p=Re(c);q=Im(c);	
 	if(p>-9999 && p<9999 &&
 //	  (fabs(y)>.034 ||x>-.9 ||fabs(x-int(x))>1.e-3) &&
 	   q>-9999 && q<9999 //&& fabs(q)> 1.e-19
 	) 
 	{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,".025 W 0 .6 0 RGB S\n");
 for(m=0;m<2;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q); fprintf(o,".025 W .9 0 0 RGB S\n");
 for(m=0;m<2;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q); fprintf(o,".025 W 0 0 .9 RGB S\n");
 for(m=1;m<15;m++)  conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".07 W .9 0 0 RGB S\n");
 for(m=1;m<15;m++)  conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".07 W 0 0 .9 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (0.  ),-9,9); fprintf(o,".07 W .6 0 .6 RGB S\n");
 for(m=-14;m<0;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".07 W 0 0 0 RGB S\n");
           m=0;     conto(o,g,w,v,X,Y,M,N, (0.+m),-9,9); fprintf(o,".07 W 0 0 0 RGB S\n");
 for(m=1;m<17;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".07 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,".02 W .5 .5 0 RGB S\n");
/*
 M(x,-8)L(x,8) fprintf(o,"0 setlinejoin 0 setlinecap 0.004 W 0 0 0 RGB S\n");
 M(x,0)L(-8.1,0) fprintf(o,"                   .05 W  1 1 1 RGB S\n");
 DO(m,23){ M(x-.4*m,0)L(x-.4*(m+.5),0);} fprintf(o,".09 W  .3 .3 0 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 facmap.eps");	
 	system(    "open facmap.pdf");	//for LINUX
 //	getchar(); system("killall Preview");//for mac
 }

Latex generator of labels

\documentclass[12pt]{article}
\usepackage{geometry}
\usepackage{graphicx}
\usepackage{rotating}
\newcommand{\sx}{\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing \includegraphics
\pagestyle{empty}
\topmargin -86pt
\oddsidemargin -96pt
\textwidth 1400pt
\textheight 1400pt
\paperwidth 1028pt
\paperheight 1010pt
\newcommand \ax {
\put( 10,342){\sx{1.4}{$y$}}
\put( 10,307){\sx{1.3}{$6$}}
\put( 10,267){\sx{1.3}{$4$}}
\put( 10,227){\sx{1.3}{$2$}}
\put( 10,187){\sx{1.3}{$0$}}
\put(  0,147){\sx{1.3}{$-2$}}
\put(  0,107){\sx{1.3}{$-4$}}
\put(  0, 67){\sx{1.3}{$-6$}}
\put(  0, 27){\sx{1.3}{$-8$}}
\put( 50, 18){\sx{1.3}{$-6$}}
\put( 90, 18){\sx{1.3}{$-4$}}
\put(130, 18){\sx{1.3}{$-2$}}
\put(178, 18){\sx{1.3}{$0$}}
\put(218, 18){\sx{1.3}{$2$}}
\put(258, 18){\sx{1.3}{$4$}}
\put(298, 18){\sx{1.3}{$6$}}
\put(334, 19){\sx{1.4}{$x$}}
}

\begin{document}
%\sx{1.2}{\begin{picture}(346,346) \ax
\sx{3}{\begin{picture}(346,346) \ax
\put(-20,-10){\includegraphics{facmap}}
\put(286,225){\rot{-7}\sx{1.6}{$v\!=\!0$}\ero}
\put(288,207){\rot{-5}\sx{1.6}{$u\!=\!0$}\ero}
\put(299,187){\sx{1.6}{$v\!=\!0$}}
\put(288,168){\rot{5}\sx{1.6}{$u\!=\!0$}\ero}
\put(286,150){\rot{7}\sx{1.6}{$v\!=\!0$}\ero}
\put(103,100){\rot{54}\sx{1.6}{$u\!=\!0$}\ero}
\put(117,100){\rot{53}\sx{1.6}{$v\!=\!0$}\ero}
\put(135,100){\rot{54}\sx{1.6}{$u\!=\!0$}\ero}
\put(152,100){\rot{54}\sx{1.6}{$v\!=\!0$}\ero}
\put(167, 99){\rot{50}\sx{1.6}{$u\!=\!0$}\ero}
\put(182, 95){\rot{41}\sx{1.6}{$v\!=\!0$}\ero}
\put(194, 85){\rot{39}\sx{1.6}{$u\!=\!0$}\ero}
\put(201, 73){\rot{35}\sx{1.6}{$v\!=\!0$}\ero}
\put(208, 61){\rot{33}\sx{1.6}{$u\!=\!0$}\ero}
\put(215, 49){\rot{28}\sx{1.6}{$v\!=\!0$}\ero}
\end{picture}}
\end{document}

References

  1. ↑ https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics - 2020/7/28
  2. ↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.
  3. ↑ https://link.springer.com/article/10.3103/S0027134910010029
    http://mizugadro.mydns.jp/PAPERS/2010superfae.pdf
    http://mizugadro.mydns.jp/PAPERS/2010superfar.pdf
    D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12. (Russian version: p.8-14)

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Date/TimeThumbnailDimensionsUserComment
current13:10, 22 August 2025Thumbnail for version as of 13:10, 22 August 20251,706 × 1,677 (932 KB)T (talk | contribs)closer to that in book Superfunctions
17:50, 20 June 2013Thumbnail for version as of 17:50, 20 June 20131,219 × 927 (479 KB)Maintenance script (talk | contribs)Importing image file

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