/// <summary>
/// Incomplete elliptic integral of the third kind Π(φ | m, n).
/// </summary>
/// <param name="φ">Argument.</param>
/// <param name="m">Parameter, equal to k², the square of the modulus.</param>
/// <param name="n" />
public static Double Π(Double φ, double m, Double n)
{
Double
sinφ = Math.Sin(φ),
c = 1 / (sinφ * sinφ);
return(CarlsonSymmetric.RF(c - 1, c - m, c) - (n / 3) * CarlsonSymmetric.RJ(c - 1, c - m, c, c + n));
}
/// <summary>
/// Complete elliptic integral of the first kind.
/// </summary>
public static double K(double m)
{
if (m == 1)
{
return(double.PositiveInfinity);
}
return(CarlsonSymmetric.RF(0, 1 - m, 1));
}
/// <summary>
/// Incomplete elliptic integral of the third kind Π(φ | m, n).
/// </summary>
/// <param name="φ">Argument.</param>
/// <param name="m">Parameter, equal to k², the square of the modulus.</param>
/// <param name="n" />
public static Complex Π(Complex φ, double m, Complex n)
{
Complex
sinφ = Complex.Sin(φ),
c = 1 / (sinφ * sinφ);
return(CarlsonSymmetric.RF(c - 1, c - m, c) - (n / 3) * CarlsonSymmetric.RJ(c - 1, c - m, c, c + n));
}
/// <summary>
/// Incomplete elliptic integral of the second kind E(φ | m).
/// </summary>
/// <param name="φ">Argument.</param>
/// <param name="m">Parameter, equal to k², the square of the modulus.</param>
public static Double E(Double φ, double m)
{
if (φ == 0)
{
return(0);
}
Double
sinφ = Math.Sin(φ),
c = 1 / (sinφ * sinφ);
return(CarlsonSymmetric.RF(c - 1, c - m, c) - (m / 3) * CarlsonSymmetric.RD(c - 1, c - m, c));
}
/// <summary>
/// Incomplete elliptic integral of the second kind E(φ | m).
/// </summary>
/// <param name="φ">Argument.</param>
/// <param name="m">Parameter, equal to k², the square of the modulus.</param>
public static Complex E(Complex φ, double m)
{
if (φ == 0)
{
return(0);
}
Complex
sinφ = Complex.Sin(φ),
c = 1 / (sinφ * sinφ);
return(CarlsonSymmetric.RF(c - 1, c - m, c) - (m / 3) * CarlsonSymmetric.RD(c - 1, c - m, c));
}
/// <summary>
/// Incomplete elliptic integral of the first kind F(φ | m).
/// </summary>
/// <param name="φ">Argument.</param>
/// <param name="m">Parameter, equal to k², the square of the modulus.</param>
public static Complex F(Complex φ, double m)
{
if (Math.Abs(φ.Real) > π / 2)
{
int n = (int)Math.Round(φ.Real / π);
return(F(φ - n * π, m) + n * 2 * K(m));
}
Complex
sinφ = Complex.Sin(φ),
cosφ = Complex.Cos(φ);
return(sinφ * CarlsonSymmetric.RF(cosφ * cosφ, 1 - m * sinφ * sinφ, 1));
}
/// <summary>
/// Incomplete elliptic integral of the first kind F(φ | m).
/// </summary>
/// <param name="φ">Argument.</param>
/// <param name="m">Parameter, equal to k², the square of the modulus.</param>
public static Double F(Double φ, double m)
{
if (Math.Abs(φ) > π / 2)
{
int n = (int)Math.Round(φ / π);
return(F(φ - n * π, m) + n * 2 * K(m));
}
Double
sinφ = Math.Sin(φ),
cosφ = Math.Cos(φ);
return(sinφ * CarlsonSymmetric.RF(cosφ * cosφ, 1 - m * sinφ * sinφ, 1));
}
/// <summary>
/// Get a function F(φ) that computes the incomplete elliptic integral of the first kind F(φ | m)
/// for a constant parameter m. Use this instead of F(φ, m) if you use the same value of m many times.
/// </summary>
/// <param name="m">Parameter, equal to k², the square of the modulus.</param>
public static Func <Complex, Complex> FComplex(double m)
{
var k = K(m);
return(φ =>
{
Complex offset = 0;
if (Math.Abs(φ.Real) > π / 2)
{
int n = (int)Math.Round(φ.Real / π);
φ -= n * π;
offset = n * 2 * k;
}
Complex
sinφ = Complex.Sin(φ),
cosφ = Complex.Cos(φ);
return sinφ * CarlsonSymmetric.RF(cosφ * cosφ, 1 - m * sinφ * sinφ, 1) + offset;
});
}
/// <summary>
/// Get a function F(φ) that computes the incomplete elliptic integral of the first kind F(φ | m)
/// for a constant parameter m. Use this instead of F(φ, m) if you use the same value of m many times.
/// </summary>
/// <param name="m">Parameter, equal to k², the square of the modulus.</param>
public static Func <Double, Double> FDouble(double m)
{
var k = K(m);
return(φ =>
{
Double offset = 0;
if (Math.Abs(φ) > π / 2)
{
int n = (int)Math.Round(φ / π);
φ -= n * π;
offset = n * 2 * k;
}
Double
sinφ = Math.Sin(φ),
cosφ = Math.Cos(φ);
return sinφ * CarlsonSymmetric.RF(cosφ * cosφ, 1 - m * sinφ * sinφ, 1) + offset;
});
}
