/// <summary>
/// Returns the complete elliptic integral of the second kind E(k) = E(π/2,k)
/// <para>E(k) = ∫ 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 EllintE(double k)
{
if (!(k >= -1 && k <= 1))
{
Policies.ReportDomainError("EllintE(k: {0}): Requires |k| <= 1", k);
return(double.NaN);
}
// Small series at k == 0
// Note that z = k^2 in the following link
// http://functions.wolfram.com/EllipticIntegrals/EllipticE/06/01/03/01/0004/
double k2 = k * k;
if (k2 < DoubleLimits.RootMachineEpsilon._13)
{
if (k2 < 4 * DoubleLimits.MachineEpsilon)
{
return(Math.PI / 2); // E(0) = Pi/2
}
return((Math.PI / 2) * HypergeometricSeries.Sum2F1(-0.5, 0.5, 1, k2));
}
if (Math.Abs(k) == 1)
{
return(1);
}
double x = 0;
double y = 1 - k2;
double z = 1;
double value = 2 * Math2.EllintRG(x, y, z);
return(value);
}