File:FactoriAsymp9ageeT.png

Map of agreement \(a_9\) for displaced Stirling asymptotic for Factorial: \[ A(z)= \sqrt{2\pi z}\ \exp\left(\log\left(\frac{z}{\mathrm e} \right) z + \frac{1}{12 z}\left(1+ \frac{1}{z^2}\left(\frac{-1}{30}+ \frac{1}{z^2}\left(\frac{1}{105}+ \frac{1}{z^2}\left(\frac{-1}{140}+ \frac{1}{z^2}\left(\frac{1}{99}+ \frac{1}{z^2}\left(\frac{-691}{30030} \right) \right) \right) \right) \right) \right) \right) \]

\[ z! \approx A_9(z)=\frac{A(z+9)}{ (z\!+\!1) (z\!+\!2) (z\!+\!3) (z\!+\!4) (z\!+\!5) (z\!+\!6) (z\!+\!7) (z\!+\!8) (z\!+\!9) } \]

The contours show the levels of agreement \[ a_9(z) = - \lg \left(\frac{|z!-A_9(z)|}{|z!|+|A_9(z)|}\right) \]

C++

/* Routines «ado.cin», «Conrec6.cin», «fac.cin» should be loaded for the compilation of the code below. Options « -std=c++11 -O2 » are strongly recommended. */

// c++ -std=c++11 FactoriAsymp9agee.cc -O2 -o FactoriAsymp9agee
#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"ado.cin"
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",1.*(x),1.*(y));
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",1.*(x),1.*(y));
#include "Conrec6.cin"
#include "fac.cin"

z_type facas(z_type z){ z_type c=1./z; z_type d=c*c;
return sqrt(M_PI*2.*z)*exp(log(z/M_E)*z+
(1./12.)*c*(
1.+d*(-1./30.+d*(1./105.+d*(-1./140.+d*(1./99.+d*(-691./30030.)))))
//1.+d*(-1./30.+d*(1./105.+d*(-1./140.+d*(1./99.+d*(-1./43.)))))
));
//(1.+c*(1./12.+c*(1./288.+c*(-139./51840.))));
}

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
int M=251,M1=M+1;
int N=201,N1=N+1;
DB X[M1],Y[N1], g[M1*N1];
//,f[M1*N1];
//FILE *o;o=fopen("FactoriAsymptoAgree.eps","w");ado(o,252,202);
FILE *o;o=fopen("FactoriAsymp9agee.eps","w");ado(o,252,202);
fprintf(o,"151 101 translate\n 10 10 scale\n");

DO(m,M1) X[m]=-15+.1*(m-.5);
DO(n,N1) Y[n]=-10+.1*(n-.5);

DO(m,M1)DO(n,N1){g[m+M1*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=Filog(z);
//c=z*z*sin(1./z);
c=fac(z);
//d=facas(z+1.)/(z+1.);
//d=facas(z+2.)/((z+1.)*(z+2.));
//d=facas(z+3.)/((z+1.)*(z+2.)*(z+3.));
//d=facas(z+4.)/((z+1.)*(z+2.)*(z+3.)*(z+4.));
//d=facas(z+5.)/((z+1.)*(z+2.)*(z+3.)*(z+4.)*(z+5.));
//d=facas(z+6.)/((z+1.)*(z+2.)*(z+3.)*(z+4.)*(z+5.)*(z+6.));
//d=facas(z+7.)/((z+1.)*(z+2.)*(z+3.)*(z+4.)*(z+5.)*(z+6.)*(z+7.));
//d=facas(z+8.)/((z+1.)*(z+2.)*(z+3.)*(z+4.)*(z+5.)*(z+6.)*(z+7.)*(z+8.));
d=facas(z+9.)/((z+1.)*(z+2.)*(z+3.)*(z+4.)*(z+5.)*(z+6.)*(z+7.)*(z+8.)*(z+9.));
p= - log( abs(c-d) / (abs(c)+abs(d)) )/log(10.);
//p=Re(c);q=Im(c); 
g[m+M1*n]=p;
       }}
fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=5.;q=1;
p=15.;
Conrec6(o,g,X,Y,M1,N1, 1. ,p); fprintf(o,".12 W 1 0 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 2. ,p); fprintf(o,".04 W 0 0 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 3. ,p); fprintf(o,".14 W .8 .7 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 4. ,p); fprintf(o,".04 W 0 0 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 5. ,p); fprintf(o,".04 W 0 0 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 6. ,p); fprintf(o,".12 W 0 1 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 7. ,p); fprintf(o,".04 W 0 0 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 8. ,p); fprintf(o,".04 W 0 0 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 9. ,p); fprintf(o,".12 W 0 .8 .9 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 10. ,p); fprintf(o,".04 W 0 0 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 11. ,p); fprintf(o,".04 W 0 0 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 12. ,p); fprintf(o,".10 W 0 0 1 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 13. ,p); fprintf(o,".04 W 0 0 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 14. ,p); fprintf(o,".04 W 0 0 0 RGB S\n");
Conrec6(o,g,X,Y,M1,N1, 15. ,p); fprintf(o,".07 W 1 .5 1 RGB S\n");

for(m=-10;m<11;m++){if(m!=0){M(m,-10)L(m,10)}}
for(n=-10;n<11;n++){M(-10,n)L(10,n)}
fprintf(o,"2 setlinecap .02 W 0 0 0 RGB S\n");
fprintf(o,"2 setlinecap .02 W 0 0 0 RGB S\n");
M(0,-10)L(0,11)
fprintf(o,"2 setlinecap .03 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
     system("epstopdf FactoriAsymp9agee.eps"); 
     system(    "open FactoriAsymp9agee.pdf"); //for mac	
}

Latex

\documentclass[12pt]{article}
\usepackage{geometry}
\paperwidth 803pt
\paperheight 669pt
\textwidth 800pt
\textheight 700pt
\topmargin -106pt
\oddsidemargin -116pt
\usepackage{graphicx}
\usepackage{rotating}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \rme {\mathrm{e}}
\newcommand \sx {\scalebox}
\begin{document}
\sx{3.15}{\begin{picture}(230,211)
\normalsize
%\put(10,8){\includegraphics{FactoriAsymp9Agree.pdf}}
\put(10,8){\includegraphics{FactoriAsymp9agee.pdf}}
\put(53,204){\sx{1.2}{\(y\)}}
\put(53,185){\sx{1.1}{\(8\)}}
\put(53,165){\sx{1.1}{\(6\)}}
\put(53,145){\sx{1.1}{\(4\)}}
\put(53,125){\sx{1.1}{\(2\)}}
\put(53,105){\sx{1.1}{\(0\)}}
\put(44,85){\sx{1.1}{\(-2\)}}
\put(44,65){\sx{1.1}{\(-4\)}}
\put(44,45){\sx{1.1}{\(-6\)}}
\put(44,25){\sx{1.1}{\(-8\)}}
\put(45, 0){\sx{.9}{\(-10\)}}
\put(73, 0){\sx{.9}{\(-8\)}}
\put(93, 0){\sx{.9}{\(-6\)}}
\put(113, 0){\sx{.9}{\(-4\)}}
\put(133, 0){\sx{.9}{\(-2\)}}
\put(159, 0){\sx{.9}{\(0\)}}
\put(179, 0){\sx{.9}{\(2\)}}
\put(199, 0){\sx{.9}{\(4\)}}
\put(219, 0){\sx{.9}{\(6\)}}
\put(239, 0){\sx{.9}{\(8\)}}
\put(256, 0){\sx{1.1}{\(x\)}}
%\put(8,194){\sx{1.4}{\rot{0}\(a\!\approx\!15\) \ero} }
%\put(132,196){\sx{1.6}{\rot{0}\(a\!\approx\!15\) \ero} }
%\put(132,176){\sx{1.6}{\rot{0}\(a\!=\!14\) \ero} }
%\put(132,156){\sx{1.6}{\rot{0}\(a\!=\!12\) \ero} }

\put( 90,50){\sx{1.6}{\rot{43}\(a_9\!=\!12\) \ero} }
\put(107,35){\sx{1.6}{\rot{44}\(a_9\!=\!14\) \ero} }
\put(122,19){\sx{1.6}{\rot{37}\(a_9\!\approx\!15\) \ero} }
\end{picture}}
\end{document} 

Keywords

«ado.cin», «Agreement», «Asymptotic», «ChatGPT», «Conrec6.cin», «Elementary function», «Factorial», «Fac.cin», «Map», «Restricted asymptotic», «Sectorial asymptotic», «Stirling»,