Difference between revisions of "File:Superfactocomple1.png"

Complex map of \(f=\)SuperFactorial(\(x\!+\!\mathrm i y\)) in the \(x,y\) plane.

Levels
\(u\!=\!\Re(f)=\)constant and
\(u\!=\!\Im(f)=\)constant are drawn.
Thick lines correspond to the integer values.

SuperFactorial, or SuFac is Superfunction of Factorial; it is solution \(F\) of the Transfer equation

\[ F(z\!+\!1)=\mathrm{Factorial} \big(F(z)\big) \] Here Factorial is treated as the Transfer function.

SuperFactorial can be expressed through the Mathematica Nest function,

SuperFactorial \((z) =\) Factorial\(^z(3) = \) Nest [ Factorial, 3, \(z\) ]

In Century 21, the Nest function is implemented only for the case when number \(z\) of the iterate can be simplified to a positive integer constant.

The complex double implementation of SuperFactorial of complex argument is described in the Moscow University Physics Bulletin, 2009 ^[1], in Russian version «Суперфункции» of book about Superfunctions, 2014 ^[2] and in the English version «Superfunctions», 2020 ^[3][4]

In particular, this map appears as the upper part of Fig.8.6 at page 96 of the Book ^[3][4]
in order to show the periodic properties of SuperFactorial.

C++ generator of curves

Files Superfactorial.cin, ado.cin, conto.cin should be loaded to the working directory in order to compile 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 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 "conto.cin"

int 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("SuperFacMap.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=superfac(z);
 //     p=abs(c-d)/(abs(c)+abs(d));  p=-log(p)/log(10.)-1.;
        p=Re(c);q=Im(c);        
        if(p>-20 && p<20 &&
 //       (fabs(y)>.034 ||x>-.9 ||fabs(x-int(x))>1.e-3) &&
           q>-20 && q<20 && 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<15;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<15;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=-14;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<17;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(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 SuperFacMap.eps");     
        system(    "open SuperFacMap.pdf");     //for LINUX
 //     getchar(); system("killall Preview");//for mac
 }

generator of labels

References

Keywords

«Factorial», «Square root of factorial», «SuFac», «SuperFactorial», «Superfunctions»,

«Корень из факториала», «Суперфункции»,