File:Lofplot.jpg

$y\!=$Lof$(x)$, thick blue line;

$y\!=$Factorial$(x)$, thin red line.

For real $x > -1~$, the simple relation takes place:

$\mathrm{Lof}(x)=\ln\!\big(\mathrm{Factorial}(x)\big)$

C++ generator of curves
files ado.cin and fac.cin should be loaded in order to compile the code below // //using namespace std; typedef std::complex z_type; //#include "facp.cin" //#include "afacc.cin" int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; FILE *o;o=fopen("lofplo.eps","w");ado(o,618,528); fprintf(o,"104 14 translate\n 100 100 scale 2 setlinecap 1 setlinejoin\n"); for(m=-1;m<6;m++){ M(m,0)L(m,5) } for(n=0;n<6;n++){ M( -1,n)L(5,n)} fprintf(o,".01 W 0 0 0 RGB S\n"); DO(n,101){ x=.025*n;x=-.82+x*x;y=Re(fac(x)); if(n==0) M(x,y) else L(x,y); if(x>0 && y>5.2) break;} fprintf(o,".012 W .8 0 0 RGB S\n"); DO(n,101){ x=.025*n;x=-.994+x*x;y=Re(lof(x)); if(n==0) M(x,y) else L(x,y);} fprintf(o,".022 W 0 0 .8 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf lofplo.eps"); system(   "open lofplo.pdf"); } //
 * 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 "ado.cin"
 * 1) define M(x,y) fprintf(o,"%6.4lf %6.5lf M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%6.4lf %6.5lf L\n",0.+x,0.+y);

Latex generator of curves
% \documentclass[12pt]{article} \usepackage{geometry} \paperwidth 612pt \paperheight 530pt \topmargin -96pt \oddsidemargin -72pt \pagestyle{empty} \usepackage{graphicx} \usepackage{rotating} \parindent 0pt \textwidth 1800px \textheight 1900px \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \begin{document} \begin{picture}(608,516) \put(-2,10){\includegraphics{lofplo}} %\put(20,10){\includegraphics{lofma}} \put(84,510){\sx{2.3}{$y$}} \put(84,415){\sx{2.3}{$4$}} \put(85,315){\sx{2.3}{$3$}} \put(86,215){\sx{2.3}{$2$}} \put(86,115){\sx{2.3}{$1$}} \put(96,1){\sx{2.3}{$0$}} \put(196,1){\sx{2.3}{$1$}} \put(296,1){\sx{2.3}{$2$}} \put(396,1){\sx{2.3}{$3$}} \put(496,1){\sx{2.3}{$4$}} \put(596,2){\sx{2.3}{$x$}} % \put(336,240){\sx{2.2}{\rot{72}$y\!=\!\mathrm{Factorial}(x)$\ero}} \put(394,206){\sx{2.4}{\rot{53}$y\!=\!\mathrm{Lof}(x)$\ero}} \end{picture} \end{document} %