/// <summary>
/// Returns the complete elliptic integral of the second kind D(k) = D(π/2,k)
/// <para>D(k) = ∫ sin^2(θ)/sqrt(1-k^2*sin^2(θ)) dθ, θ={0,π/2}</para>
/// </summary>
/// <param name="k">The modulus. Requires |k| ≤ 1</param>
public static double EllintD(double k)
{
if (!(k >= -1 && k <= 1))
{
Policies.ReportDomainError("EllintD(k: {0}): Requires |k| <= 1", k);
return(double.NaN);
}
// special values
if (k == 0)
{
return(Math.PI / 4);
}
if (Math.Abs(k) == 1)
{
return(Double.PositiveInfinity);
}
// http://dlmf.nist.gov/19.5#E3
if (k * k < DoubleLimits.RootMachineEpsilon._13)
{
return((Math.PI / 4) * HypergeometricSeries.Sum2F1(1.5, 0.5, 2, k * k));
}
double x = 0;
double y = 1 - k * k;
double z = 1;
double value = Math2.EllintRD(x, y, z) / 3;
return(value);
}
// 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 E(φ, k)
/// <para>E(φ, k) = ∫ 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 EllintE(double k, double phi)
{
if ((!(k >= -1 && k <= 1)) || (double.IsNaN(phi)))
{
Policies.ReportDomainError("EllintE(k: {0}, phi: {1}): Requires |k| <= 1; phi not NaN", k, phi);
return(double.NaN);
}
if (double.IsInfinity(phi))
{
return(phi);
}
if (Math.Abs(phi) > Trig.PiReductionLimit)
{
Policies.ReportNotImplementedError("EllintE(k: {0}, phi: {1}): |phi| > {2} not implemented", k, phi, Trig.PiReductionLimit);
return(double.NaN);
}
// special values
if (k == 0)
{
return(phi);
}
if (phi == 0)
{
return(0);
}
if (phi == Math.PI / 2)
{
return(Math2.EllintE(k));
}
// Carlson's algorithm works only for |phi| <= π/2,
// use the integrand's periodicity to normalize phi
// E(k, phi + π*mult) = E(k, phi) + 2*mult*E(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 * EllintE(k);
}
}
if (k == 1)
{
result += Math.Sin(rphi);
}
else
{
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;
result += sinp * (Math2.EllintRF(x, y, z) - t * Math2.EllintRD(x, y, z) / 3);
}
return((phi < 0) ? -result : result);
}
