/// <summary>
/// Returns the Laczos approximation of Γ(z)
/// </summary>
public static double Tgamma(double x)
{
Debug.Assert(x > 0);
// check for large x to avoid potential Inf/Inf below
if (x > DoubleLimits.MaxGamma + 1)
{
return(double.PositiveInfinity);
}
// use the Lanczos approximation
double xgh = x + (Lanczos.G - 0.5);
double result = Lanczos.Series(x);
if (x > MaxPowerTermX)
{
// we're going to overflow unless this is done with care:
double hp = Math.Pow(xgh, (x / 2) - 0.25);
result *= (hp / Math.Exp(xgh));
result *= hp;
}
else
{
result *= (Math.Pow(xgh, x - 0.5) / Math.Exp(xgh));
}
return(result);
}
/// <summary>
/// Returns Γ(z)/Γ(z+delta) using the Lanczos approximation
/// </summary>
public static double TgammaDeltaRatio(double x, double delta)
{
Debug.Assert(x > 0 && delta + x > 0);
// Compute Γ(x)/Γ(x+Δ), which reduces to:
//
// ((x+g-0.5)/(x+Δ+g-0.5))^(x-0.5) * (x+Δ+g-0.5)^-Δ * e^Δ * (Sum(x)/Sum(x+Δ))
// = (1+Δ/(x+g-0.5))^-(x-0.5) * (x+Δ+g-0.5)^-Δ * e^Δ * (Sum(x)/Sum(x+Δ))
double xgh = x + (Lanczos.G - 0.5);
double xdgh = x + delta + (Lanczos.G - 0.5);
double mu = delta / xgh;
double result = Lanczos.Series(x) / Lanczos.Series(x + delta);
result *= (Math.Abs(mu) < 0.5) ? Math.Exp((0.5 - x) * Math2.Log1p(mu)) : Math.Pow(xgh / xdgh, x - 0.5);
result *= Math.Pow(Math.E / xdgh, delta);
return(result);
}
