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
using namespace std; typedef complex z_type; //#include "sinc.cin"
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "fac.cin"
 * 1) include "facp.cin"
 * 2) include "afacc.cin"
 * 3) include "superfactorial.cin"
 * 4) include "abelfac.cin"
 * 5) 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 % % Copyleft 2012 by Dmitrii Kouznetsov % \documentclass[12pt]{article} % \usepackage{geometry} % \usepackage{graphicx} % \usepackage{rotating} % % \paperwidth 340pt % % \paperheight 336pt % \paperwidth 680pt % \paperheight 672pt % %\topmargin -96pt % %\oddsidemargin -98pt % \topmargin -104pt % \oddsidemargin -128pt % \textwidth 1100pt % \textheight 1100pt % \pagestyle {empty} % \newcommand \sx {\scalebox} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \ing {\includegraphics} % \parindent 0pt \pagestyle{empty} \begin{document}% \sx{2}{ \begin{picture}(362,362) % \put(0,0){\ing{QFacMap}} % \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{1.3}{$-8$}} % \put(70,30){\sx{1.3}{$-6$}} % \put(110,30){\sx{1.3}{$-4$}} % \put(150,30){\sx{1.3}{$-2$}} % \put(198,30){\sx{1.3}{$0$}} % \put(238,30){\sx{1.3}{$2$}} % \put(278,30){\sx{1.3}{$4$}} % \put(318,30){\sx{1.3}{$6$}} % \put(354,30){\sx{1.3}{$x$}} % \put(70,198){\sx{1.3}{\bf cut}} %central% \put(47,162){\sx{1.3}{\rot{72}$u\!=\!0$\ero}} % \put(99,161){\sx{1.3}{\rot{80}$v\!=\!0$\ero}} % \put(139,162){\sx{1.3}{\rot{89}$u\!=\!0$\ero}} % \put(164,204){\sx{1.3}{\rot{77}$v\!=\!0$\ero}} % \put(177,200){\sx{1.2}{\rot{56}$u\!=\!0$\ero}} % % \put(78,139){\sx{1.1}{\rot{0.}$u\!=\!0.2$\ero}} % \put(82,116){\sx{1.1}{\rot{0.}$u\!=\!0.4$\ero}} % \put(86,102){\sx{1.1}{\rot{0.}$u\!=\!0.6$\ero}} % % \put(128,144){\sx{1.1}{\rot{0.}$v\!=\!0.2$\ero}} % \put(137,128){\sx{1.1}{\rot{0.}$v\!=\!0.4$\ero}} % % % \put(318,294){\sx{1.3}{\rot{4}$u\!=\!-8$\ero}} % \put(332,252){\sx{1.3}{\rot{23}$u\!=\!8$\ero}} % % \put(331,220){\sx{1.3}{\rot{-12}$v\!=\!8$\ero}} % %%\put(337,198){\sx{1.1}{\rot{0.}$v\!=\!0$\ero}} % \put(321,174){\sx{1.3}{\rot{9}$v\!=\!-8$\ero}} % % \put(326,146){\sx{1.3}{\rot{-28}$u\!=\!8$\ero}} % \put(319,102){\sx{1.3}{\rot{-7}$u\!=\!-8$\ero}} % \end{picture} % } % \end{document} % %