File:QFacMapT.png

Complex map of function Square root of factorial, $\sqrt{\,!\!}$, id est

$f(z)=\mathrm{Factorial}^{1/2}(z)$

$u\!+\!\mathrm i v=f(x\!+\!\mathrm i y)$

This function is implemented through the SuperFactorial and AbelFactorial in the following way:

$f(z)= \mathrm{SuperFactorial}\Big(0.5+\mathrm{AbelFactorial}(z)\Big)$

In wide range of values $z$,

$ f(f(z))=z!$

C++ generator of curves

#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 "superfactorial.cin"
#include "abelfac.cin"
#include "conto.cin"

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("QFacMap.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);
//     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 QFacMap.eps"); 
       system(    "open QFacMap.pdf"); //for LINUX
//     getchar(); system("killall Preview");//for mac
}

Latex generator of labels

% file QFacMap.pdf should be generated with the code above in order to compile the Latex document below. % % Gerenator of QFacMapT.png %<br> % Copyleft 2012 by Dmitrii Kouznetsov %<br> \documentclass[12pt]{article} %<br> \usepackage{geometry} %<br> \usepackage{graphicx} %<br> \usepackage{rotating} %<br> % \paperwidth 340pt %<br> % \paperheight 336pt %<br> \paperwidth 680pt %<br> \paperheight 672pt %<br> %\topmargin -96pt %<br> %\oddsidemargin -98pt %<br> \topmargin -104pt %<br> \oddsidemargin -128pt %<br> \textwidth 1100pt %<br> \textheight 1100pt %<br> \pagestyle {empty} %<br> \newcommand \sx {\scalebox} %<br> \newcommand \rot {\begin{rotate}} %<br> \newcommand \ero {\end{rotate}} %<br> \newcommand \ing {\includegraphics} %<br> \parindent 0pt \pagestyle{empty} \begin{document}%<br> \sx{2}{ \begin{picture}(362,362) %<br> \put(0,0){\ing{QFacMap}} %<br> \put(30,357){\sx{1.3}{$y$}} %<br> \put(30,317){\sx{1.3}{$6$}} %<br> \put(30,277){\sx{1.3}{$4$}} %<br> \put(30,237){\sx{1.3}{$2$}} %<br> \put(29,196){\sx{1.3}{$0$}} %<br> \put(20,156){\sx{1.3}{$-2$}} %<br> \put(20,116){\sx{1.3}{$-4$}} %<br> \put(20,76){\sx{1.3}{$-6$}} %<br> \put(20,36){\sx{1.3}{$-8$}} %<br> \put(70,30){\sx{1.3}{$-6$}} %<br> \put(110,30){\sx{1.3}{$-4$}} %<br> \put(150,30){\sx{1.3}{$-2$}} %<br> \put(198,30){\sx{1.3}{$0$}} %<br> \put(238,30){\sx{1.3}{$2$}} %<br> \put(278,30){\sx{1.3}{$4$}} %<br> \put(318,30){\sx{1.3}{$6$}} %<br> \put(354,30){\sx{1.3}{$x$}} %<br> \put(70,198){\sx{1.3}{\bf cut}} %central%<br> \put(47,162){\sx{1.3}{\rot{72}$u\!=\!0$\ero}} %<br> \put(99,161){\sx{1.3}{\rot{80}$v\!=\!0$\ero}} %<br> \put(139,162){\sx{1.3}{\rot{89}$u\!=\!0$\ero}} %<br> \put(164,204){\sx{1.3}{\rot{77}$v\!=\!0$\ero}} %<br> \put(177,200){\sx{1.2}{\rot{56}$u\!=\!0$\ero}} %<br> % \put(78,139){\sx{1.1}{\rot{0.}$u\!=\!0.2$\ero}} %<br> \put(82,116){\sx{1.1}{\rot{0.}$u\!=\!0.4$\ero}} %<br> \put(86,102){\sx{1.1}{\rot{0.}$u\!=\!0.6$\ero}} %<br> % \put(128,144){\sx{1.1}{\rot{0.}$v\!=\!0.2$\ero}} %<br> \put(137,128){\sx{1.1}{\rot{0.}$v\!=\!0.4$\ero}} %<br> % % \put(318,294){\sx{1.3}{\rot{4}$u\!=\!-8$\ero}} %<br> \put(332,252){\sx{1.3}{\rot{23}$u\!=\!8$\ero}} %<br> % \put(331,220){\sx{1.3}{\rot{-12}$v\!=\!8$\ero}} %<br> %%\put(337,198){\sx{1.1}{\rot{0.}$v\!=\!0$\ero}} %<br> \put(321,174){\sx{1.3}{\rot{9}$v\!=\!-8$\ero}} %<br> % \put(326,146){\sx{1.3}{\rot{-28}$u\!=\!8$\ero}} %<br> \put(319,102){\sx{1.3}{\rot{-7}$u\!=\!-8$\ero}} %<br> \end{picture} %<br> } %<br> \end{document} %<br> %