/// <summary>
/// Computes the digamma function.
/// </summary>
/// <param name="x">The argument.</param>
/// <returns>The value of ψ(x).</returns>
/// <remarks>
/// <para>The psi function, also called the digamma function, is the logrithmic derivative of the Γ function.</para>
/// <img src="../images/DiGamma.png" />
/// <para>To evaluate the Psi function for complex arguments, use <see cref="AdvancedComplexMath.Psi" />.</para>
/// </remarks>
/// <seealso cref="Gamma(double)"/>
/// <seealso cref="AdvancedComplexMath.Psi"/>
/// <seealso href="http://en.wikipedia.org/wiki/Digamma_function" />
/// <seealso href="http://mathworld.wolfram.com/DigammaFunction.html" />
public static double Psi(double x)
{
if (x <= 0.0)
{
if (x == Math.Ceiling(x))
{
// there are poles at zero and negative integers
return(Double.NaN);
}
else
{
// use the reflection formula to change to a positive x
return(Psi(1.0 - x) - Math.PI / Math.Tan(Math.PI * x));
}
}
else if (x < 16.0)
{
return(Lanczos.Psi(x));
}
else
{
// for large arguments, the Stirling asymptotic expansion is faster than the Lanzcos approximation
return(Stirling.Psi(x));
//return (Psi_Stirling(x));
}
}
/// <summary>
/// Computes the digamma function.
/// </summary>
/// <param name="x">The argument.</param>
/// <returns>The value of ψ(x).</returns>
/// <remarks>
/// <para>The psi function, also called the digamma function, is the logrithmic derivative of the Γ function.</para>
/// <img src="../images/DiGamma.png" />
/// <para>Because it is defined as a <i>logarithmic</i> derivative, the digamma function does not overflow <see cref="System.Double"/>
/// even for arguments for which <see cref="Gamma(double)"/> does.</para>
/// <para>To evaluate the psi function for complex arguments, use <see cref="AdvancedComplexMath.Psi" />.</para>
/// </remarks>
/// <seealso cref="Gamma(double)"/>
/// <seealso cref="AdvancedComplexMath.Psi"/>
/// <seealso href="http://en.wikipedia.org/wiki/Digamma_function" />
/// <seealso href="http://mathworld.wolfram.com/DigammaFunction.html" />
public static double Psi(double x)
{
if (x < 0.5)
{
return(Psi(1.0 - x) - Math.PI / MoreMath.TanPi(x));
}
else if (x < 1.5)
{
return(GammaSeries.PsiOnePlus(x - 1.0));
}
else if (x < 2.5)
{
return(GammaSeries.PsiTwoPlus(x - 2.0));
}
else if (x < 16.0)
{
return(Lanczos.Psi(x));
}
else if (x <= Double.PositiveInfinity)
{
// For large arguments, the Stirling asymptotic expansion is faster than the Lanzcos approximation
return(Stirling.Psi(x));
}
else
{
return(Double.NaN);
}
}
