// 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);
}
// Elliptic integrals (complete and incomplete) of the third kind
// Carlson, Numerische Mathematik, vol 33, 1 (1979)
public static double Imp(double k, double n, double nc)
{
// Note arg nc = 1-n, possibly without cancellation errors
Debug.Assert(k >= -1 && k <= 1, "Requires |k| <= 1: k = " + k);
// special values
// See: http://dlmf.nist.gov/19.6.E3
if (n == 0)
{
if (k == 0)
{
return(Math.PI / 2);
}
return(Math2.EllintK(k));
}
// special values
// See: http://functions.wolfram.com/EllipticIntegrals/EllipticPi/03/01/01/
if (k == 0)
{
return((Math.PI / 2) / Math.Sqrt(nc));
}
if (k == 1)
{
return(double.NegativeInfinity / Math.Sign(n - 1));
}
if (n < 0)
{
// when the magnitude of n gets very large, N below becomes ~= 1, so
// there can be some error, so use the series at n=Infinity
// http://functions.wolfram.com/EllipticIntegrals/EllipticPi/06/01/05/
if (n < -1 / DoubleLimits.MachineEpsilon)
{
return((Math.PI / 2) / Math.Sqrt(-n) + (Math2.EllintE(k) - Math2.EllintK(k)) / n);
}
// A&S 17.7.17:
// shift to k^2 < N < 1
double k2 = k * k;
double N = (k2 - n) / nc;
double Nc = (1 - k2) / nc; // Nc = 1-N
double result = k2 / (k2 - n) * Math2.EllintK(k);
result -= (n / nc) * ((1 - k2) / (k2 - n)) * _EllintPi.Imp(k, N, Nc);
return(result);
}
double x = 0;
double y = 1 - k * k;
double z = 1;
double p = nc;
double value = Math2.EllintK(k) + n * Math2.EllintRJ(x, y, z, p) / 3;
return(value);
}
