/// <summary>
/// Calculates the associated legendre polynomial(m,l) with parameter x.
/// </summary>
/// <param name="l"></param>
/// <param name="m"></param>
/// <param name="x"></param>
/// <returns></returns>
public static double AssociatedPolynomial(int l, int m, double x)
{
Contract.Requires(-1 <= x && x <= 1);
if (l < 0)
{
return(AssociatedPolynomial(-l - 1, m, x));
}
if (m < 0)
{
int sign = Functions.IsOdd(m) ? 1 : -1;
return(sign * Functions.GammaRatio(l + m + 1, l + 1 - m) * AssociatedPolynomial(l, -m, x));
}
if (m > l)
{
return(0);
}
if (m == 0)
{
return(Polynomial(l, x));
}
double p0 = Functions.Factorial(Functions.Factorial(2 * m - 1));
if (Functions.IsOdd(m))
{
p0 *= -1;
}
if (m == l)
{
return(p0);
}
double p1 = x * (2 * m + 1) * p0;
for (int n = m + 1; n < l; ++n)
{
var p2 = Next(n, m, x, p0, p1);
p0 = p1;
p1 = p2;
}
return(p1);
}