File:QfacMapT500a.jpg

Complex map of function [[square root of factorial, or $\sqrt{!}$, or halfiteration of factorial

$h=\mathrm{Factorial}^{1/2}$

is such function that $ h(h(z)) = z!$ in wide range of values of $z$.

Levels $u=\Re(h(z))=\rm const$ and Levels $u=\Im(h(z))=\rm const$ are shown in the plane $z=x+\mathrm i y$.

C++ generator of curves
// In order to compile the code below, the following files should be loaded in the working directory: // fac.cin , // facp.cin , // afacc.cin , // superfac.cin , // arcsuperfac.cin , // ado.cin , // conto.cin // using namespace std; typedef complex z_type; //#include "superex.cin" //#include "superlo.cin" DB xL=0.31813150520476413; DB yL=1.3372357014306895;
 * 1) include 
 * 2) include 
 * 3) include 
 * 1) include
 * 2) define DB double
 * 3) define DO(x,y) for(x=0;x<y;x++)
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "fac.cin"
 * 5) include "facp.cin"
 * 6) include "afacc.cin"
 * 7) include "superfac.cin"
 * 8) include "arcsuperfac.cin"


 * 1) include "conto.cin"

int main{ int j,k,m,n,n1; 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("qfacMap.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,N1){  y=-8.+.04*n; if(y<-.011) Y[n]=y; else break;} Y[n]= -.01; n++; Y[n]= +.01; n++; for(j=n;j-999 && p<999)    g[m*N1+n]=p; if(q>-999 && q<999 && fabs(q)> 1.e-14) f[m*N1+n]=q; }} fprintf(o,"1 setlinejoin 1 setlinecap\n"); //p=6.;q=6.; //#include"plofu.cin" p=1;q=.5; for(m=-2;m<2;m++) for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M-20,N,(m+.1*n),-q, q); fprintf(o,".02 W 0 .6 0 RGB S\n"); for(m= 0;m<2;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M-20,N,-(m+.1*n),-q, q); fprintf(o,".02 W .9 0 0 RGB S\n"); for(m= 0;m<2;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M-20,N, (m+.1*n),-q, q); fprintf(o,".02 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,".04 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,".04 W 0 0 .9 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".04 W .6 0 .6 RGB S\n"); for(m=-4;m<5;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".04 W 0 0 0 RGB S\n"); fprintf(o,"0 setlinejoin 0 setlinecap\n"); x=.85; M(x,0)L(-8.1,0)                                fprintf(o,".07 W 1 1 1 RGB S\n"); DO(m,22){ M(x-.4*m,0) L(x-.4*(m+.5),0) }       fprintf(o,".1 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf qfacMap.eps"); system(   "open qfacMap.pdf");         // for mac //     getchar; system("killall Preview");   //for macintosh //return 0; }

Latex generator of labels
% \documentclass[12pt]{article} \usepackage{graphicx} \usepackage{geometry} \paperwidth 340pt \paperheight 340pt \usepackage{rotating} \textwidth 800mm \textheight 400mm \topmargin -61mm \oddsidemargin -77pt \parindent 0pt \newcommand \sx {\scalebox} % \newcommand \rot \begin{rotate} % % \newcommand \ero \end{rotate} % \newcommand \ax { \put( 10,346){\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, 17){\sx{1.3}{$-6$}} \put( 90, 17){\sx{1.3}{$-4$}} \put(130, 17){\sx{1.3}{$-2$}} \put(178, 17){\sx{1.3}{$0$}} \put(218, 17){\sx{1.3}{$2$}} \put(258, 17){\sx{1.3}{$4$}} \put(298, 17){\sx{1.3}{$6$}} \put(334, 17){\sx{1.4}{$x$}} } \begin {document} \begin{picture}(350,420) %\put(-20,-10){\includegraphics{fig4a}}\ax \put(-20,-10){\includegraphics{qfacMap}}\ax \put(20,174){\sx{1.1}{$u$=0}} \put(66,160){\sx{1.1}{$v$=0}} \put(101,174){\sx{1.1}{$u$=0}} \put(132,174){\sx{1.1}{$v$=0}} {\put(70,187){\sx{1.8}{\bf cut}}} \put(300,277){\rot{ 10}\sx{1.3}{$u\!=\!-4$}\ero} \put(310,250){\rot{16}\sx{1.3}{$u\!=  4$}\ero} \put(300,205){\rot{-7}\sx{1.3}{$v\!=  4$}\ero} \put(300,172){\rot{ 6}\sx{1.3}{$v\!=\!-4$}\ero} \put(308,128){\rot{-23}\sx{1.3}{$u\!=  4$}\ero} \put(300, 99){\rot{-15}\sx{1.3}{$u\!=\!-4$}\ero} \end{picture} \end{document} %