/// <summary>
/// Computes the given harmonic number.
/// </summary>
/// <param name="n">The index of the harmonic number to compute, which must be non-negative.</param>
/// <returns>The harmonic number H<sub>n</sub>.</returns>
/// <exception cref="ArgumentOutOfRangeException"><paramref name="n"/> is negative.</exception>
/// <remarks>
/// <para>H<sub>n</sub> is the nth partial sum of the harmonic series.</para>
/// <img src="..\images\HarmonicSeries.png" />
/// <para>Since the harmonic series diverges, H<sub>n</sub> grows without bound as n increases, but
/// it does so extremely slowly, approximately as log(n).</para>
/// </remarks>
/// <exception cref="ArgumentOutOfRangeException"><paramref name="n"/> is negative.</exception>
/// <seealso href="http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)"/>
public static double HarmonicNumber(int n)
{
if (n < 0)
{
throw new ArgumentOutOfRangeException(nameof(n));
}
else if (n < 32)
{
// for small values, just add up the harmonic series
double H = 0.0;
for (int i = 1; i <= n; i++)
{
H += 1.0 / i;
}
return(H);
}
else
{
// for large values, use the digamma function
// this will route to the the Stirling asymptotic expansion
return(AdvancedMath.Psi(n + 1) + AdvancedMath.EulerGamma);
}
}