File:AbelFactorialMap.png

Complex map of AbelFactorial.

Function $f=\mathrm{AbelFactorial}(x+\mathrm i y)$ is shown with lines

$\Re(f)\!=$const anf lines $\Im(f)\!=$ const

C++ Generator of curves
The original generator is misplaced, but the code below does very similar figure

Files fac.cin, facp.cin, afacc.cin, superfactorial.cin, abelfac.cin, conto.cin should be loaded to the working directory un order to compile the C++ code below. 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"

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("AbelFacMap.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); c=abelfac(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=-5;m<5;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(2,0)L(-8.1,0) fprintf(o,"0 setlinejoin 0 setlinecap .05 W  1 1 1 RGB S\n"); DO(m,25){ M(2-.4*(m+.2),0)L(2-.4*(m+.4),0);} fprintf(o,".09 W 1 .4 0 RGB S\n"); DO(m,25){ M(2-.4*(m+.7),0)L(2-.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 AbelFacMap.eps"); system(   "open AbelFacMap.pdf");      //for LINUX //    getchar; system("killall Preview");//for mac }

Latex Generator of labels
The original generator is misplaced, but the code below does very similar figure \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{AbelFacMap}} \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(50,306){\sx{1.3}{$u\!=\!1.8$}} \put(50,255){\sx{1.3}{$v\!=\!0.6$}} \put(90,198){\sx{1.3}{\bf cut}} %central \put(50,150){\sx{1.3}{$v\!=\!-0.6$}} \put(48,90){\sx{1.3}{$u\!=\!1.8$}} % column \put(170,344){\sx{1.3}{$u\!=\!1.6$}} %\put(250,314){\sx{1.3}{$u\!=\!1.2$}} \put(119,230){\sx{1.3}{$v\!=\!0.8$}} \put(109,170){\sx{1.3}{$v\!=\!-\!0.8$}} \put(144,090){\sx{1.3}{$u\!=\!1.6$}} \put(252,084){\sx{1.3}{$u\!=\!1.4$}} % column \put(319,344){\sx{1.3}{$u\!=\!1.4$}} \put(304,306){\sx{1.3}{$v\!=\!0.4$}} \put(313,278){\sx{1.3}{$u\!=\!1.2$}} \put(309,236){\sx{1.3}{$v\!=\!0.2$}} %\put(266,247){\sx{1.}{$u\!=\!0.8856031944$}} \put(306,209){\sx{1.3}{$u\!=\!1$}} \put(331,198){\sx{1.3}{$v\!=\!0$}} %central \put(312,118){\sx{1.3}{$u\!=\!1.2$}} \put(304,90){\sx{1.3}{$v\!=\!-0.4$}} \put(320, 50){\sx{1.3}{$u\!=\!1.4$}} \end{picture} } \end{document}