// Elliptic integrals (complete and incomplete) of the second kind
// Carlson, Numerische Mathematik, vol 33, 1 (1979)
/// <summary>
/// Returns the incomplete elliptic integral of the second kind D(φ, k)
/// <para>D(φ, k) = ∫ sin^2(θ)/sqrt(1-k^2*sin^2(θ)) dθ, θ={0,φ}</para>
/// </summary>
/// <param name="k">The modulus. Requires |k| ≤ 1</param>
/// <param name="phi">The amplitude</param>
public static double EllintD(double k, double phi)
{
if ((!(k >= -1 && k <= 1)) || (double.IsNaN(phi)))
{
Policies.ReportDomainError("EllintD(k: {0}, phi: {1}): Requires |k| <= 1; phi not NaN", k, phi);
return(double.NaN);
}
if (double.IsInfinity(phi))
{
return(phi); // Note: result != phi, only +/- infinity
}
if (Math.Abs(phi) > Trig.PiReductionLimit)
{
Policies.ReportNotImplementedError("EllintD(k: {0}, phi: {1}): |phi| > {2} not implemented", k, phi, Trig.PiReductionLimit);
return(double.NaN);
}
// special values
//if (k == 0) // too much cancellation error near phi=0, general case works fine
// return (phi - Cos(phi)*Sin(phi))/2
if (phi == 0)
{
return(0);
}
if (phi == Math.PI / 2)
{
return(Math2.EllintD(k));
}
// Carlson's algorithm works only for |phi| <= π/2,
// use the integrand's periodicity to normalize phi
// D(k, phi + π*mult) = D(k, phi) + 2*mult*D(k)
double result = 0;
double rphi = Math.Abs(phi);
if (rphi > Math.PI / 2)
{
// Normalize periodicity so that |rphi| <= π/2
var(angleMultiple, angleRemainder) = Trig.RangeReducePI(rphi);
double mult = 2 * angleMultiple;
rphi = angleRemainder;
if (mult != 0)
{
result += mult * EllintD(k);
}
}
double k2 = k * k;
double sinp = Math.Sin(rphi);
double cosp = Math.Cos(rphi);
double x = cosp * cosp;
double t = k2 * sinp * sinp;
double y = (t < 0.875) ? 1 - t : (1 - k2) + k2 * x;
double z = 1;
// http://dlmf.nist.gov/19.25#E13
// and RD(lambda*x, lambda*y, lambda*z) = lambda^(-3/2) * RD(x, y, z)
result += EllintRD(x, y, z) * sinp * sinp * sinp / 3;
return((phi < 0) ? -result : result);
}
