public static ComplexDouble FastGamma(ComplexDouble n)
{
if (ComplexDouble.IsNaN(n))
{
if (ThrowExceptions)
{
throw new ArgumentException("n is NaN", "n");
}
else
{
return(double.NaN);
}
}
if (n.Im.Equals(0d))
{
return(FastGamma(n.Re));
}
if (ComplexDouble.IsInfinity(n))
{
return(0d);
}
var q = new ComplexDouble(n.Re % 1d, n.Im);
if (q.Re < 0d)
{
++q;
}
var z = (int)(n.Re - q.Re);
n = q + 8d;
return(q.Equals(0d) || q.Equals(1d) ? Pochhammer(q + 1d, z - 1) :
q.Equals(0.5d) ? 0.5d * ComplexDouble.SqrtPi * Pochhammer(q + 1d, z - 1) :
ComplexDouble.Exp(
0.5d * System.Math.Log(2d * ComplexDouble.Pi)
+ (n - 0.5d) * ComplexDouble.Log(n) - n
+ Pow(n, -1) / 12d
- Pow(n, -3) / 360d
+ Pow(n, -5) / 1260d
- Pow(n, -7) / 1680d
+ Pow(n, -9) / 1188d
- Pow(n, -11) / 360360d * 691d
+ Pow(n, -13) / 156d
) * Pochhammer(n, z - 8));
/*
* n = q + 17d;
* return
* ComplexDouble.Pow(2d * ComplexDouble.Pi * n, 0.5d) * ComplexDouble.Pow(n, n - 1d)
* ComplexDouble.Exp(-n + 1d / (12d * n) - 1d / (360d * ComplexDouble.Pow(n, 3d))
+ 1d / (1260d * ComplexDouble.Pow(n, 5d)) - 1d / (1686d * ComplexDouble.Pow(n, 7d)))
* Pochhammer(n, z - 17);
* //*/
}
public static ComplexDouble IntegrateComplex(Func <double, ComplexDouble> f, double a, double b, ComplexDouble fa, ComplexDouble fb, Func2 <double> nextPoint, double eps = Eps)
{
bool ia = ComplexDouble.IsInfinity(fa) || ComplexDouble.IsNaN(fa),
ib = ComplexDouble.IsInfinity(fb) || ComplexDouble.IsNaN(fb);
if ((ia || ib) && eps <= Eps)
{
return(0d);
}
double p = nextPoint(a, b);
ComplexDouble fp = f(p),
ab = 0.5d * (b - a) * (fa + fb),
ap = 0.5d * (p - a) * (fa + fp),
pb = 0.5d * (b - p) * (fp + fb),
s = ap + pb;
if ((ab - s).SqrAbs <= eps * eps * (b - a) * (b - a))
{
return(s);
}
return(IntegrateComplex(f, a, p, fa, fp, nextPoint, eps)
+ IntegrateComplex(f, p, b, fp, fb, nextPoint, eps)); //*/
}
public static ComplexDouble SlowGamma(ComplexDouble n, double eps = Eps)
{
if (ComplexDouble.IsNaN(n))
{
if (ThrowExceptions)
{
throw new ArgumentException("n is NaN", "n");
}
else
{
return(double.NaN);
}
}
if (n.Im.Equals(0d))
{
return(SlowGamma(n.Re, eps));
}
if (ComplexDouble.IsInfinity(n))
{
return(0d);
}
var q = new ComplexDouble(n.Re % 1d, n.Im);
if (q.Re < 0d)
{
++q;
}
var z = (int)(n.Re - q.Re);
n = new ComplexDouble(q.Re + q.Re + 1d, q.Im + q.Im);
Func <double, ComplexDouble> f = t => ComplexDouble.Exp(n * System.Math.Log(t) - t * t);
return(2d * (
IntegrateComplex(f, 0d, 1d, 0d, 1d / ComplexDouble.E, eps) +
IntegrateComplex(f, 1d, 26.641747557046328d, 1d / ComplexDouble.E, 0d, eps)
) * Pochhammer(++q, --z));
}
public static ComplexDouble HyperSum2F1(ComplexDouble a, ComplexDouble b, ComplexDouble c, ComplexDouble z, Predicate <ComplexDouble> pred, bool thrw = true)
{
if (ComplexDouble.IsNaN(a))
{
if (thrw)
{
throw new ArgumentOutOfRangeException("a", "a is NaN");
}
else
{
return(a);
}
}
if (ComplexDouble.IsNaN(b))
{
if (thrw)
{
throw new ArgumentOutOfRangeException("b", "b is NaN");
}
else
{
return(b);
}
}
if (ComplexDouble.IsNaN(c))
{
if (thrw)
{
throw new ArgumentOutOfRangeException("c", "c is NaN");
}
else
{
return(c);
}
}
if (ComplexDouble.IsNaN(z))
{
if (thrw)
{
throw new ArgumentOutOfRangeException("z", "z is NaN");
}
else
{
return(z);
}
}
if (c.Im.Equals(0d) && c.Re <= 0d && (c.Re % 1d).Equals(0d))
{
if (thrw)
{
throw new ArgumentOutOfRangeException("c", "c is 0 or negative integer");
}
else
{
return(double.PositiveInfinity);
}
}
double i = z.SqrAbs;
if (i > 1d || i.Equals(1d) && (c - a - b).Re <= 0d)
{
if (thrw)
{
throw new ArgumentOutOfRangeException("z", "|z| >= 1");
}
else
{
return(double.PositiveInfinity);
}
}
i = 0d;
ComplexDouble prod = 1d, sum = 0d;
while (pred(prod))
{
sum += prod *= (z * a++ *b++) / (++i * c++);
}
return(sum + 1d);
}