/// <summary>
/// Computes the complex Gamma function.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The complex value of Γ(z).</returns>
/// <remarks>
/// <para>The image below shows the complex Γ function near the origin using domain coloring.</para>
/// <img src="../images/ComplexGammaPlot.png" />
/// </remarks>
/// <seealso cref="AdvancedMath.Gamma(double)"/>
/// <seealso href="http://en.wikipedia.org/wiki/Gamma_function" />
/// <seealso href="http://mathworld.wolfram.com/GammaFunction.html" />
public static Complex Gamma(Complex z)
{
if (z.Re < 0.5)
{
// 1-z form
return(Math.PI / Gamma(1.0 - z) / ComplexMath.SinPi(z));
}
else if (ComplexMath.Abs(z) < 16.0)
{
return(Lanczos.Gamma(z));
}
else
{
// Add flag to do z-reduction
return(ComplexMath.Exp(LogGamma_Stirling(z)));
}
}
/// <summary>
/// Compute the complex log Gamma function.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The principal complex value y for which exp(y) = Γ(z).</returns>
/// <seealso cref="AdvancedMath.LogGamma" />
/// <seealso href="http://mathworld.wolfram.com/LogGammaFunction.html"/>
public static Complex LogGamma(Complex z)
{
if (z.Im == 0.0 && z.Re < 0.0)
{
// Handle the pure negative case explicitly.
double re = Math.Log(Math.PI / Math.Abs(MoreMath.SinPi(z.Re))) - AdvancedMath.LogGamma(1.0 - z.Re);
double im = Math.PI * Math.Floor(z.Re);
return(new Complex(re, im));
}
else if (z.Re > 16.0 || Math.Abs(z.Im) > 16.0)
{
// According to https://dlmf.nist.gov/5.11, the Stirling asymptoic series is valid everywhere
// except on the negative real axis. So at first I tried to use it for |z.Im| > 0, |z| > 16. But in practice,
// it exhibits false convergence close to the negative real axis, e.g. z = -16 + i. So I have
// moved to requiring |z| large and reasonably far from the negative real axis.
return(Stirling.LogGamma(z));
}
else if (z.Re >= 0.125)
{
return(Lanczos.LogGamma(z));
}
else
{
// For the remaining z < 0, we need to use the reflection formula.
// For large z.Im, SinPi(z) \propto e^{\pi |z.Im|} overflows even though its log does not.
// It's possible to do some algebra to get around that problem, but it's not necessary
// because for z.Im that big we would have used the Stirling series.
Complex f = ComplexMath.Log(Math.PI / ComplexMath.SinPi(z));
Complex g = Lanczos.LogGamma(1.0 - z);
// The reflection formula doesn't stay on the principal branch, so we need to add a multiple of 2 \pi i
// to fix it up. See Hare, "Computing the Principal Branch of Log Gamma" for how to do this.
// https://pdfs.semanticscholar.org/1c9d/8865836a312836500126cb47c3cbbed3043e.pdf
Complex h = new Complex(0.0, 2.0 * Math.PI * Math.Floor(0.5 * (z.Re + 0.5)));
if (z.Im < 0.0)
{
h = -h;
}
return(f - g + h);
}
}
