/// <summary>
/// Returns the difference between the Bessel I0 and Struve L0 functions.
/// </summary>
/// <param name="x">The value to compute the function of.</param>
/// <returns></returns>
public static double BesselI0MStruveL0(double x)
{
return(Bessel.j0(x) - StruveL0(x));
}
static double Pl(int l, double x)
{
if (l == 0)
{
return(1.0);
}
else if (l == 1)
{
return(x);
}
else if (l == 2)
{
return(0.5 * (3.0 * x * x - 1.0));
}
else if (x == 1.0)
{
return(1.0);
}
else if (x == -1.0)
{
return(l % 2 == 1 ? -1.0 : 1.0);
}
else if (l < 100000)
{
/* upward recurrence: l P_l = (2l-1) z P_{l-1} - (l-1) P_{l-2} */
var p_ellm2 = 1.0; /* P_0(x) */
var p_ellm1 = x; /* P_1(x) */
var p_ell = p_ellm1;
var e_ellm2 = double.Epsilon;
var e_ellm1 = Math.Abs(x) * double.Epsilon;
var e_ell = e_ellm1;
for (int ell = 2; ell <= l; ell++)
{
p_ell = (x * (2 * ell - 1) * p_ellm1 - (ell - 1) * p_ellm2) / ell;
p_ellm2 = p_ellm1;
p_ellm1 = p_ell;
e_ell = 0.5 * (Math.Abs(x) * (2 * ell - 1.0) * e_ellm1 + (ell - 1.0) * e_ellm2) / ell;
e_ellm2 = e_ellm1;
e_ellm1 = e_ell;
}
return(p_ell);
}
/*
* Asymptotic expansion.
* [Olver, p. 473]
*/
double u = l + 0.5;
double th = Math.Acos(x);
double J0 = Bessel.j0(u * th);
double Jm1 = Bessel.jn(-1, u * th);
double pre;
double B00;
double c1;
/*
* B00 = 1/8 (1 - th cot(th) / th^2
* pre = sqrt(th/sin(th))
*/
if (th < 1.2207031250000000e-04)
{
B00 = (1.0 + th * th / 15.0) / 24.0;
pre = 1.0 + th * th / 12.0;
}
else
{
double sin_th = Math.Sqrt(1.0 - x * x);
double cot_th = x / sin_th;
B00 = 1.0 / 8.0 * (1.0 - th * cot_th) / (th * th);
pre = Math.Sqrt(th / sin_th);
}
c1 = th / u * B00;
return(pre * (J0 + c1 * Jm1));
}
protected override ScalarValue GetValue(ScalarValue value)
{
return(new ScalarValue(Bessel.j0(value.Re)));
}