File:AfacmapT800.png

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\[ u+\mathrm iv = \mathrm{ArcFactorial}(x\!+\!\mathrm i y) \]

This map is used as Fig.8.4 at page 93 of book «Superfunctions»^[1]^[2]
in order to remind the well-known special function; but also as a test for the complex double numeric implementation of ArcFactorial.

The similar map appears also in article about Square root of factorial ^[3].
No precedent complex double implementation of ArcFactorial is found at the free access sites.

In publications mentioned, Factorial is considered as the Transfer function. For this function, the Superfunction (SuFac) and the Abelfunction (AuFac) are constructed with Regular iteration at the Fixed point \(L=2\). Then the \(n\)th iterate of Factorial is expressed as follows:

\[ \mathrm{Factorial}^n(z)=\mathrm{SuFac}\big( n + \mathrm{AuFac}(z)\big) \]

For the imlementation of SuFac and AuFac, the ArcFactorial (shown in the map) needs to be implemented. Plotting the complex map is important part of testing of the numerical implementation of a holomorphic function.

C++ generator of curves

fac.cin

// The code below should be stored as fac.cin

z_type fracti(z_type z){ z_type s; int n;  DB a[17]=
{0.0833333333333333333, 0.0333333333333333333,  .252380952380952381, .525606469002695418, 
1.01152306812684171,   1.51747364915328740,   2.26948897420495996, 3.00991738325939817, 
4.02688719234390123,   5.00276808075403005,   6.28391137081578218, 7.49591912238403393, 
9.04066023436772670,  10.4893036545094823,   12.2971936103862059, 13.9828769539924302, 
16.0535514167049355 };
s=a[16]/(z+19./(z+25./(z)));  for(n=15;n>=0;n--) s=a[n]/(z+s);
return  s + log(2.*M_PI)/2. - z + (z+.5)*log(z);
} //  logfactorial for large values of argument except vicinity of negative part of real axis)

z_type infac0(z_type z){ z_type s; int n;  DB c[28]={ 1.,
 0.57721566490153286061,        -0.65587807152025388108,
-0.042002635034095235529,        0.16653861138229148950,
-0.042197734555544336748,       -0.0096219715278769735621,
 0.0072189432466630995424,      -0.0011651675918590651121,
-0.00021524167411495097282,      0.00012805028238811618615,
-0.000020134854780788238656,    -0.0000012504934821426706573,
 0.0000011330272319816958824,   -2.0563384169776071035e-7,
 6.1160951044814158179e-9,       5.0020076444692229301e-9,
-1.1812745704870201446e-9,       1.0434267116911005105e-10,
 7.7822634399050712540e-12,     -3.6968056186422057082e-12,
 5.1003702874544759790e-13,     -2.0583260535665067832e-14,
-5.3481225394230179824e-15,      1.2267786282382607902e-15,
-1.1812593016974587695e-16,      1.1866922547516003326e-18,
 1.4123806553180317816e-18};
 s=c[27]*z; for(n=26;n>0;n--) {s+=c[n]; s*=z;}
 s+=c[0];  return s;}

z_type fac0(z_type z){ return 1./infac0(z);}

z_type expaun(z_type z) {int n,m; DB x,y;
               x=Re(z);if(x<-.5) return expaun(z+1.)-log(z+1.);
                       if(x>.6) return expaun(z-1.)+log(z);
               y=Im(z); if(fabs(y)>1.4)return expaun(z/2.)+expaun(z/2.-.5)+z*log(2.)-log(sqrt(M_PI));
                       return -log(infac0(z)); }

z_type lof(z_type z){DB x,y; x=Re(z); y=Im(z);
        if(fabs(y)>5. ) return fracti(z);
        if(x>0 && x*x+y*y>25.) return fracti(z);
       return expaun(z);  } // lof(z) returns 16 digits of complex logfactorial.

 z_type infac1(z_type z){return infac0(z/2.)*infac0((z-1.)/2.)*sqrt(M_PI)/exp(log(2.)*z);}
 z_type infac2(z_type z){return infac1(z/2.)*infac1((z-1.)/2.)*sqrt(M_PI)/exp(log(2.)*z);}
 z_type infac3(z_type z){return infac2(z/2.)*infac2((z-1.)/2.)*sqrt(M_PI)/exp(log(2.)*z);}
 z_type inhalf(z_type z){DB x=Re(z); DB y=Im(z); DB r=x*x+y*y;
                       if(r<2.) return infac0(z); 
                       if(r<5.) return infac1(z);
                                return infac2(z); }

z_type infacmi(z_type z){ if(Re(z)> 1.) return infacmi(z-1.)/z;     return inhalf(z);}
z_type infaclu(z_type z){ if(Re(z)<-.5) return infaclu(z+1.)*(z+1.);return inhalf(z);}

z_type infac(z_type z){DB x=Re(z),y=Im(z),t=x*x+y*y; if(t<1.)return infac0(z);
       if( fabs(y)> 5. || (x>0 && t>25) )              return exp(-fracti(z));
       if( x>0 ) return infacmi(z);
                 return infaclu(z);}

z_type fac(z_type z){ DB x=Re(z),y=Im(z),t=x*x+y*y; if(t<2.)return 1./infac0(z);
               if( (x>0. && t>25.) || fabs(y)>5.)         return exp(fracti(z));
               if(x>0) return 1./infacmi(z);
                       return 1./infaclu(z);}

afacc.cin

// The code below should be stored as afacc.cin

z_type afacb(z_type z){
DB z0=0.461632144968362341262659542325721328468196204;
DB F0=-0.12148629053584960809551455717769158215135617313;
DB c2=.483836122723810585213722380854825370205628608;
DB p=0.2090973242496979633924701135209125815611056;
DB q=0.0565790271828431799463572817754001404669620;
DB A=0.0008685913050832152753870514845664790993724;
DB B=0.0002046727298252365296379380008904113017495;
z_type t=(log(z)-F0)/c2;   z_type v=sqrt(t);
z_type u=v*(1.+v*(p+A*t))
         /(1.+v*(q+B*t)) + z0; return u;}
z_type afacc(z_type z){ z_type a,c,d;   a=afacb(z); 
d=facp(a); c=z-fac(a); a+=c/d; if(abs(c)<1.e-12) return a;
d=facp(a); c=z-fac(a); a+=c/d; if(abs(c)<1.e-12) return a;
d=facp(a); c=z-fac(a); a+=c/d; if(abs(c)<1.e-12) return a;
d=facp(a); c=z-fac(a); a+=c/d; if(abs(c)<1.e-12) return a;
d=facp(a); c=z-fac(a); a+=c/d; return a;
}

Main

Files fac.cin, afacc.cin above and ado.cin and conto.cin should be loaded in the working directory for compillation of the C++ 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 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=400,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("afacmap.eps","w");ado(o,402,402);
fprintf(o,"201 201 translate\n 20 20 scale\n");
DO(m,M1) X[m]=-8.+.04*(m);
DO(n,200)Y[n]=-8.+.04*n;
        Y[200]=-.01;
        Y[201]= .01;
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>-6.9 && p<6.9 &&
//       (fabs(y)>.034 ||x>-.9 ||fabs(x-int(x))>1.e-3) &&
          q>-6.9 && q<6.9 //&& 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=.4;q=.4;
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<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,".025 W 0 0 .9 RGB S\n");
for(m=1;m<5;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<5;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.  ),-p,p); fprintf(o,".07 W .6 0 .6 RGB S\n");
for(m=-4;m<7;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;
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 afacmap.eps"); 
       system(    "open afacmap.pdf"); //for LINUX
//     getchar(); system("killall Preview");//for mac
}

Latex generator of labels

% File afacmap.pdf should be generated with the code above in order to compile the Latex document below:

%

 %<br>
\documentclass[12pt]{article} %<br>
\usepackage{geometry} %<br>
\usepackage{graphicx} %<br>
\usepackage{rotating} %<br>
\usepackage{hyperref} %<br> 
\paperwidth 339px %<br>
\paperheight 336px %<br>
\textwidth 165mm %<br>
\textheight 240mm %<br>
\topmargin -96pt %<br>
\oddsidemargin -76pt %<br>
\parindent 0pt %<br>
\begin {document} %<br>
\newcommand \sx {\scalebox} %<br>
\newcommand \rme {{e}}	 %<br>
\newcommand \rmi {{\rm i}}	%imaginary unity is always roman font %<br>
\newcommand \ds {\displaystyle} %<br>
\newcommand \bN {\mathbb{N}} %<br>
\newcommand \bC {\mathbb{C}} %<br>
\newcommand \bR {\mathbb{R}} %<br>
\newcommand \cO {\mathcal{O}} %<br>
\newcommand \cF {\mathcal{F}} %<br>
\newcommand \rot {\begin{rotate}} %<br>
\newcommand \ero {\end{rotate}} %<br>
\newcommand \nS {\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!} %<br>
\newcommand \pS {{~}~{~}} %<br>
\newcommand \fac {\mathrm{Factorial}} %<br>
\newcommand \ax { %<br>
\put( 10,342){\sx{1.4}{$y$}} %<br>
\put( 10,307){\sx{1.3}{$6$}} %<br>
\put( 10,267){\sx{1.3}{$4$}} %<br>
\put( 10,227){\sx{1.3}{$2$}} %<br>
\put( 10,187){\sx{1.3}{$0$}} %<br>
\put(  0,147){\sx{1.3}{$-2$}} %<br>
\put(  0,107){\sx{1.3}{$-4$}} %<br>
\put(  0, 67){\sx{1.3}{$-6$}} %<br>
\put(  0, 27){\sx{1.3}{$-8$}} %<br>
\put( 50, 18){\sx{1.3}{$-6$}} %<br>
\put( 90, 18){\sx{1.3}{$-4$}} %<br>
\put(130, 18){\sx{1.3}{$-2$}} %<br>
\put(178, 18){\sx{1.3}{$0$}} %<br>
\put(218, 18){\sx{1.3}{$2$}} %<br>
\put(258, 18){\sx{1.3}{$4$}} %<br>
\put(298, 18){\sx{1.3}{$6$}} %<br>
\put(334, 19){\sx{1.4}{$x$}} %<br>
} %<br>
\begin{picture}(340,340) \ax %<br>
\put(-20,-10){\includegraphics{afacmap}} %<br>
\put(158,348){\rot{-73}\sx{1.5}{$q\!=\!1.2$}\ero} %<br>
\put(102,346){\rot{-51}\sx{1.5}{$q\!=\!1.4$}\ero} %<br>
\put( 61,313){\rot{-34}\sx{1.5}{$q\!=\!1.6$}\ero} %<br>
\put( 38,268){\rot{-16}\sx{1.5}{$q\!=\!1.8$}\ero} %<br>
\put( 32,236){\rot{ -2}\sx{1.5}{$q\!=\!2$}\ero} %<br>
\put( 32,188){\sx{1.8}{\bf cut}} %<br>
\put( 33,152){\sx{1.5}{$q\!=\!-2$}} %<br>
\put( 36,106){\rot{15}\sx{1.5}{$q\!=\!-1.8$}\ero} %<br>
\put( 64, 60){\rot{35}\sx{1.5}{$q\!=\!-1.6$}\ero} %<br>
\put(112, 33){\rot{53}\sx{1.5}{$q\!=\!-1.4$}\ero} %<br>
\put(166, 27){\rot{74}\sx{1.5}{$q\!=\!-1.2$}\ero} %<br>
% %<br>
\put(225,308){\rot{82}\sx{1.5}{$q\!=\!1$}\ero} %<br>
\put(219, 83){\rot{-87}\sx{1.5}{$q\!=\!-1$}\ero} %<br>
\put(244, 89){\rot{-68}\sx{1.5}{$q\!=\!-0.8$}\ero} %<br>
\put(253,293){\rot{62}\sx{1.5}{$q\!=\!0.8$}\ero} %<br>
\put(280,276){\rot{45}\sx{1.5}{$q\!=\!0.6$}\ero} %<br>
\put(300,250){\rot{30}\sx{1.5}{$q\!=\!0.4$}\ero} %<br>
\put(298,226){\rot{15}\sx{1.5}{$q\!=\!0.2$}\ero} %<br>
\put(304,187){\sx{1.6}{$q$=0}} %<br>
%\put(300,162){\rot{-15}\sx{1.5}{$q\!=\!-0.2$}\ero} %<br>
\put(264,132){\rot{54}\sx{1.5}{$p$=2.8}\ero} %<br>
\put(280,110){\rot{50}\sx{1.5}{$p$=3}\ero} %<br>
\put(290, 70){\rot{44}\sx{1.5}{$p$=3.2}\ero} %<br>
\put(300, 42){\rot{39}\sx{1.5}{$p$=3.4}\ero} %<br>
\end{picture} %<br>
\end{document} %<br>
%

References

↑ 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
↑ 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.
↑ https://link.springer.com/article/10.3103/S0027134910010029
http://mizugadro.mydns.jp/PAPERS/2010superfae.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)

Keywords

«ArcFactorial», «Factorial», «Gamma function», «Table of superfunctions», «Transfer function», «Superfunctions»,

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[1] ttps://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] ttps://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] ttps://link.springer.com/article/10.3103/S0027134910010029
http://mizugadro.mydns.jp/PAPERS/2010superfae.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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