/// <summary>
/// Returns Iv(x) for small <paramref name="x"/>
/// </summary>
/// <param name="v"></param>
/// <param name="x"></param>
/// <returns></returns>
public static double I_SmallArg(double v, double x)
{
double prefix;
if (v <= DoubleLimits.MaxGamma - 1) {
prefix = Math.Pow(x / 2, v) / Math2.Tgamma(v + 1);
} else {
prefix = StirlingGamma.PowDivGamma(x / 2, v) / v;
}
if (prefix == 0)
return prefix;
double series = HypergeometricSeries.Sum0F1(v + 1, x * x / 4);
return prefix * series;
}
/// <summary>
/// Series evaluation for Jv(x), for small x
/// </summary>
/// <param name="v"></param>
/// <param name="x"></param>
/// <returns></returns>
public static double J_SmallArg(double v, double x)
{
Debug.Assert(v >= 0);
// See http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/06/01/04/01/01/0003/
// Converges rapidly for all x << v.
// Most of the error occurs calculating the prefix
double prefix;
if (v <= DoubleLimits.MaxGamma - 1)
prefix = Math.Pow(x / 2, v) / Math2.Tgamma(v + 1);
else
prefix = StirlingGamma.PowDivGamma(x / 2, v)/v;
if (prefix == 0)
return prefix;
double series = HypergeometricSeries.Sum0F1(v + 1, -x * x / 4);
return prefix * series;
}
