ArcGamma.jpg/code

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//
(CC) Image: Dmitrii Kouznetsov
// generator of picture of ArcFactorial in the complex plane. 
// Copyleft 2009 by Dmitrii Kouznetsov.
// The image and the generator can be used for free under condition that the authorship and the source are attributed.
// For the plotting, two more routines are required;
// get them at Contour plot/Code/conto.cin
//   
#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.)

z_type fracti(z_type z){ z_type s; int n;  DB a[17]=
{0.0833333333333333333, 0.0333333333333333333,  .252380952380952381, .525606469002695418, 
1.01152306812684171,   1.51747364915328740,   2.26948897420495996, 3.00991738325939817, 
4.02688719234390123,   5.00276808075403005,   6.28391137081578218, 7.49591912238403393, 
9.04066023436772670,  10.4893036545094823,   12.2971936103862059, 13.9828769539924302, 
16.0535514167049355 };
s=a[16]/(z+19./(z+25./(z)));  for(n=15;n>=0;n--) s=a[n]/(z+s);
return  s + log(2.*M_PI)/2. - z + (z+.5)*log(z);
} //  logfactorial for large values of argument except vicinity of negative part of real axis)
z_type infac0(z_type z){ z_type s; int n;  DB c[28]={ 1.,
 0.57721566490153286061,        -0.65587807152025388108,
-0.042002635034095235529,        0.16653861138229148950,
-0.042197734555544336748,       -0.0096219715278769735621,
 0.0072189432466630995424,      -0.0011651675918590651121,
-0.00021524167411495097282,      0.00012805028238811618615,
-0.000020134854780788238656,    -0.0000012504934821426706573,
 0.0000011330272319816958824,   -2.0563384169776071035e-7,
 6.1160951044814158179e-9,       5.0020076444692229301e-9,
-1.1812745704870201446e-9,       1.0434267116911005105e-10,
 7.7822634399050712540e-12,     -3.6968056186422057082e-12,
 5.1003702874544759790e-13,     -2.0583260535665067832e-14,
-5.3481225394230179824e-15,      1.2267786282382607902e-15,
-1.1812593016974587695e-16,      1.1866922547516003326e-18,
 1.4123806553180317816e-18};
 s=c[27]*z; for(n=26;n>0;n--) {s+=c[n]; s*=z;}
 s+=c[0];  return s;}
z_type fac0(z_type z){ return 1./infac0(z);}
z_type expaun(z_type z) {int n,m; DB x,y;
		x=Re(z);if(x<-.5) return expaun(z+1.)-log(z+1.);
			if(x>.6) return expaun(z-1.)+log(z);
		y=Im(z); if(fabs(y)>1.4)return expaun(z/2.)+expaun(z/2.-.5)+z*log(2.)-log(sqrt(M_PI));
			return -log(infac0(z)); }

z_type lof(z_type z){DB x,y; x=Re(z); y=Im(z);
        if(fabs(y)>5. ) return fracti(z);
        if(x>0 && x*x+y*y>25.) return fracti(z);
        return expaun(z);  } // lof(z) returns 16 digits of complex logfactorial.
  z_type infac1(z_type z){return infac0(z/2.)*infac0((z-1.)/2.)*sqrt(M_PI)/exp(log(2.)*z);}
  z_type infac2(z_type z){return infac1(z/2.)*infac1((z-1.)/2.)*sqrt(M_PI)/exp(log(2.)*z);}
  z_type infac3(z_type z){return infac2(z/2.)*infac2((z-1.)/2.)*sqrt(M_PI)/exp(log(2.)*z);}
  z_type inhalf(z_type z){DB x=Re(z); DB y=Im(z); DB r=x*x+y*y;
			if(r<2.) return infac0(z); 
			if(r<5.) return infac1(z);
			         return infac2(z); }
z_type infacmi(z_type z){ if(Re(z)> 1.) return infacmi(z-1.)/z;     return inhalf(z);}
z_type infaclu(z_type z){ if(Re(z)<-.5) return infaclu(z+1.)*(z+1.);return inhalf(z);}
z_type infac(z_type z){DB x=Re(z),y=Im(z),t=x*x+y*y; if(t<1.)return infac0(z);
	if( fabs(y)> 5. || (x>0 && t>25) )              return exp(-fracti(z));
	if( x>0 ) return infacmi(z);
	          return infaclu(z);}

z_type fac(z_type z){ DB x=Re(z),y=Im(z),t=x*x+y*y; if(t<2.)return 1./infac0(z);
		if( (x>0. && t>25.) || fabs(y)>5.)         return exp(fracti(z));
		if(x>0) return 1./infacmi(z);
		        return 1./infaclu(z);}
//#include "fac.cin"
// Functions for Factorial'(z)
z_type infp0(z_type z)
{ int n; z_type s; DB c[30]= 
{0.57721566490153286061,	-1.3117561430405077622,
-0.12600790510228570659,	 0.66615444552916595801,
-0.21098867277772168374,	-0.057731829167261841373,
 0.050532602726641696797,	-0.0093213407348725208969,
-0.0019371750670345587553,	 0.0012805028238811618615,
-0.00022148340258867062521,	-0.000015005921785712047888,
 0.000014729354015762046471,	-2.8788737837686499448e-6,
 9.1741426567221237268e-8,	 8.0032122311507566881e-8,
-2.0081667698279342458e-8,	 1.8781680810439809189e-9,
 1.4786300535819635383e-10,	-7.3936112372844114164e-11,
 1.0710777603654399556e-11,	-4.5283173178463149231e-13,
-1.2300681840672941359e-13,	 2.9442687077718258964e-14,
-2.9531482542436469238e-15,	 3.0853998623541608647e-17,
 3.8134277693586858102e-17,	-6.4364879164190365785e-18,
 4.9717783335892785568e-19,	 4.0120551914810793446e-21};
s=c[29]; for(n=28;n>=0;n--){ s*=z; s+=c[n];}   return s;}
z_type lofp0(z_type z){ return -infp0(z)/infac(z); }
z_type facp0(z_type z){ z_type f=fac(z); f*=f;  return -infp0(z)*f; }

z_type lofpa(z_type z){ int n; DB q[11];  q[0] = 12.; q[1] = 5./6.;  q[2] = 252./79.;  
q[3] = 6241./14460.; q[4 ]= 7666692./4146631.; q[5 ]= 179081182865./612465549066.;
q[6 ]= 4881681043696812./3754087889491759.;
q[7 ]= 86960333299682003491937./392729697097736725384440.;
q[8 ]= 378191910699307315313565647105916./377413323237205130354503096392253.;
q[9 ]= 696148976661357653747206985359295786942014225./
       3903889440300118372577892204070110729027524454.;
q[10]= 36675782764501469367480729990524142326314524131790623634298644./
       45019657243089322180478800624616560743983830599801241354133773.;
z_type c=1./(z*z),   s=c/q[10];   for(n=9;n>=0;n--) s=c/(q[n]+s);
return -s + .5/z + log(z);}

	z_type lofp2(z_type z){ return log(2.)+(lofp0(z/2.-.5)+lofp0(z/2.))/2.;}
	z_type lofp4(z_type z){ return log(2.)+(lofp2(z/2.-.5)+lofp2(z/2.))/2.;}
	z_type lofp8(z_type z){ return log(2.)+(lofp4(z/2.-.5)+lofp4(z/2.))/2.;}
z_type lofp1(z_type z){DB x=Re(z),y=Im(z), t=x*x+y*y; 
		if(x>1) return lofp1(z-1.)+1./z;
		if(x<-.5) return lofp1(z+1.)-1./(z+1.);
		if(t<2.) return lofp0(z); 
		return lofp4(z); }
z_type lofp(z_type z){DB x=Re(z),y=Im(z), u=y*y;
if(x>=0. && (x+1.)*(x+1.)+u> 30.)return lofpa(z);
if(x<=0. && (x-1.)*(x-1.)+u> 30.)return lofpa(-z)+1./z-M_PI/tan(M_PI*z);
return lofp1(z);}  
z_type infp(z_type z){ DB x=Re(z),y=Im(z);
		if(x*x+y*y<2.)	return infp0(z);
				return -infac(z)*lofp(z);}
z_type facp(z_type z){ DB x=Re(z),y=Im(z), u=x*x+y*y; z_type c;
if(u<2){c=infac0(z);	return -infp0(z)/(c*c);}
	return fac(z)*lofp(z);
//if(x>0|| fabs(y)>2.)return M_PI*insincp(M_PI*z)*infp(-z)-insinc(M_PI*z)*infp(-z);
}
// #include "facp.cin"
z_type afacb(z_type z){
DB z0=0.461632144968362341262659542325721328468196204;
DB F0=-0.12148629053584960809551455717769158215135617313;
DB c2=.483836122723810585213722380854825370205628608;
DB p=0.2090973242496979633924701135209125815611056;
DB q=0.0565790271828431799463572817754001404669620;
DB A=0.0008685913050832152753870514845664790993724;
DB B=0.0002046727298252365296379380008904113017495;
z_type t=(log(z)-F0)/c2;   z_type v=sqrt(t);
z_type u=v*(1.+v*(p+A*t))
	   /(1.+v*(q+B*t)) + z0; return u;}
z_type afacc(z_type z){ z_type a,c,d;   a=afacb(z); 
d=facp(a); c=z-fac(a); a+=c/d; if(abs(c)<1.e-12) return a;
d=facp(a); c=z-fac(a); a+=c/d; if(abs(c)<1.e-12) return a;
d=facp(a); c=z-fac(a); a+=c/d; if(abs(c)<1.e-12) return a;
d=facp(a); c=z-fac(a); a+=c/d; if(abs(c)<1.e-12) return a;
d=facp(a); c=z-fac(a); a+=c/d;	return a;
}
//#include "afacc.cin"
#include "conto.cin"
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
  int M=201,M1=M+1;
  int N=201,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("ArcFactorial.eps","w");ado(o,202,202);
fprintf(o,"101 101 translate\n 10 10 scale\n");
DO(m,M1) X[m]=-10.+.1*(m-.5);
DO(n,N1) Y[n]=-10.+.1*(n+.5);
for(m=-8;m<9;m++) {	if(m==0){M(m,-6.2)L(m,6.2)} else	{M(m,-6)L(m,6)}			}
for(n=-6;n<7;n++) {	M(  -8,n)L(8,n)}
fprintf(o,".006 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);
//	p=abs(c-d)/(abs(c)+abs(d));  p=-log(p)/log(10.)-1.;
	p=Re(c);q=Im(c);	
	if(p>-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=2;q=1;
for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-4.+.1*n),-q, q); fprintf(o,".01 W 0 1 0 RGB S\n");
for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-3.+.1*n),-q, q); fprintf(o,".01 W 0 1 0 RGB S\n");
for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-2.+.1*n),-q, q); fprintf(o,".01 W 0 1 0 RGB S\n");
for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (-1.+.1*n),-q, q); fprintf(o,".01 W 0 1 0 RGB S\n");
for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, ( 0.+.1*n),-q, q); fprintf(o,".01 W 0 1 0 RGB S\n");
for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, ( 1.+.1*n),-q, q); fprintf(o,".01 W 0 1 0 RGB S\n");
for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, ( 2.+.1*n),-q, q); fprintf(o,".01 W 0 1 0 RGB S\n");
for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, ( 3.+.1*n),-q, q); fprintf(o,".01 W 0 1 0 RGB S\n");
for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (-4.+.1*n),-q, q); fprintf(o,".01 W 1 0 0 RGB S\n");
for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (-3.+.1*n),-q, q); fprintf(o,".01 W 1 0 0 RGB S\n");
for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (-2.+.1*n),-q, q); fprintf(o,".01 W 1 0 0 RGB S\n");
for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (-1.+.1*n),-q, q); fprintf(o,".01 W 1 0 0 RGB S\n");
for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, ( 0.+.1*n),-q, q); fprintf(o,".01 W 0 0 1 RGB S\n");
for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, ( 1.+.1*n),-q, q); fprintf(o,".01 W 0 0 1 RGB S\n");
for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, ( 2.+.1*n),-q, q); fprintf(o,".01 W 0 0 1 RGB S\n");
for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, ( 3.+.1*n),-q, q); fprintf(o,".01 W 0 0 1 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (-8.     ),-p,p); fprintf(o,".020 W 1 0 0 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (-7.     ),-p,p); fprintf(o,".020 W 1 0 0 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (-6.     ),-p,p); fprintf(o,".020 W 1 0 0 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (-5.     ),-p,p); fprintf(o,".020 W 1 0 0 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (-4.     ),-p,p); fprintf(o,".020 W 1 0 0 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (-3.     ),-p,p); fprintf(o,".020 W 1 0 0 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (-2.     ),-p,p); fprintf(o,".020 W 1 0 0 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, (-1.     ),-p,p); fprintf(o,".020 W 1 0 0 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, ( 0.     ),-p,p); fprintf(o,".020 W 1 0 1 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, ( 1.     ),-p,p); fprintf(o,".020 W 0 0 1 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, ( 2.     ),-p,p); fprintf(o,".020 W 0 0 1 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, ( 3.     ),-p,p); fprintf(o,".020 W 0 0 1 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, ( 4.     ),-p,p); fprintf(o,".020 W 0 0 1 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, ( 5.     ),-p,p); fprintf(o,".020 W 0 0 1 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, ( 6.     ),-p,p); fprintf(o,".020 W 0 0 1 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, ( 7.     ),-p,p); fprintf(o,".020 W 0 0 1 RGB S\n");
                  conto(o,f,w,v,X,Y,M,N, ( 8.     ),-p,p); fprintf(o,".020 W 0 0 1 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, (-8.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, (-7.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, (-6.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, (-5.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, (-4.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, (-3.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, (-2.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, (-1.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, ( 0.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, ( 1.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, ( 2.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, ( 3.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, ( 4.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, ( 5.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, ( 6.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, ( 7.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
                 conto(o,g,w,v,X,Y,M,N, ( 8.     ),-p,p); fprintf(o,".020 W 0 0 0 RGB S\n");
//#include"plofu.cin"
M(.885,0)L(-10,0) fprintf(o,".06 W 1 1 1 RGB S\n");
M(.885,0)L(-10,0) fprintf(o,".05 W 0 0 0 RGB [.12 .12] 1 setdash  S\n");
fprintf(o,"showpage\n\%\%\%Trailer"); fclose(o);
//	system(    "gv ArcFactorial.eps &"); // for linux
	system(  "open ArcFactorial.eps"); // for macintosh
	system("ps2pdf ArcFactorial.eps $"); 
	getchar(); system("killall Preview"); //for macintosh
}
//end of generator of arcfactorial