// Elliptic integrals (complete and incomplete) of the first kind
// Carlson, Numerische Mathematik, vol 33, 1 (1979)
/// <summary>
/// Returns the incomplete elliptic integral of the first kind
/// <para>F(φ, k) = F(φ | k^2) = F(sin(φ); k)</para>
/// <para>F(φ, k) = ∫ dθ/sqrt(1-k^2*sin^2(θ)), θ={0,φ}</para>
/// <para>F(x; k) = ∫ dt/sqrt((1-t^2)*(1-k^2*t^2)), t={0,x}</para>
/// </summary>
/// <param name="k">The modulus. Requires |k| ≤ 1</param>
/// <param name="phi">The amplitude</param>
public static double EllintF(double k, double phi)
{
if ((!(k >= -1 && k <= 1)) || (double.IsNaN(phi)))
{
Policies.ReportDomainError("EllintF(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("EllintF(k: {0}, phi: {1}): |phi| > {2} not implemented", k, phi, Trig.PiReductionLimit);
return(double.NaN);
}
#if EXTRA_DEBUG
Debug.WriteLine("EllintF(phi = {0}; k = {1})", phi, k);
#endif
// special values
if (k == 0)
{
return(phi);
}
if (phi == 0)
{
return(0);
}
if (phi == Math.PI / 2)
{
return(Math2.EllintK(k));
}
// Carlson's algorithm works only for |phi| <= π/2,
// use the integrand's periodicity to normalize phi
// F(k, phi + π*mult) = F(k, phi) + 2*mult*K(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 * EllintK(k);
}
}
double sinp = Math.Sin(rphi);
double cosp = Math.Cos(rphi);
double k2 = k * k;
double x = cosp * cosp;
double y = 1 - k2 * sinp * sinp;
if (y > 0.125) // use a more accurate form with less cancellation
{
y = (1 - k2) + k2 * x;
}
double z = 1;
result += sinp * EllintRF(x, y, z);
return((phi < 0) ? -result : result);
}
public static double Imp(double k, double n, double nc, double phi)
{
// Note arg nc = 1-n, possibly without cancellation errors
Debug.Assert(k >= -1 && k <= 1, "Requires |k| <= 1: k = " + k);
// See: http://dlmf.nist.gov/19.6#iv
if (phi == 0)
{
return(0);
}
if (phi == Math.PI / 2)
{
return(Math2.EllintPi(k, n));
}
if (n == 0)
{
if (k == 0)
{
return(phi);
}
return(Math2.EllintF(k, phi));
}
// Carlson's algorithm works only for |phi| <= π/2,
// use the integrand's periodicity to normalize phi
// Π(k, n, phi + π*mult) = Π(k, n, phi) + 2*mult*Π(k,n)
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) && (nc > 0))
{
result = mult * _EllintPi.Imp(k, n, nc);
}
}
if (k == 0)
{
double ncr;
// A&S 17.7.20:
if (n < 1)
{
if (nc == 1)
{
return(phi);
}
ncr = Math.Sqrt(nc);
result += Math.Atan(ncr * Math.Tan(rphi)) / ncr;
}
else if (n == 1)
{
result += Math.Tan(rphi);
}
else
{
// n > 1:
ncr = Math.Sqrt(-nc);
result += Math2.Atanh(ncr * Math.Tan(rphi)) / ncr;
}
return((phi < 0) ? -result : result);
}
double sphi = Math.Sin(Math.Abs(phi));
if (n > 1 / (sphi * sphi))
{
// Complex result is a domain error:
Policies.ReportDomainError("EllintPi(k: {0}, nu: {1}, phi: {2}): Complex results not supported. Requires n > 1 / sin^2(phi)", k, n, phi);
return(double.NaN);
}
// Special cases first:
if (n < 0)
{
//
// If we don't shift to 0 <= n <= 1 we get
// cancellation errors later on. Use
// A&S 17.7.15/16 to shift to n > 0:
//
double k2 = k * k;
double N = (k2 - n) / nc;
double Nc = (1 - k2) / nc; // Nc = 1-N = (1-k^2)/nc
// check to see if a rounding error occurred
// k^2 <= N <= 1
if (N < k2)
{
#if EXTRA_DEBUG
Debug.WriteLine("Rounding error: EllintPi(k: {0} , nu: {1} , phi: {2})", k, n, phi);
#endif
N = k2;
Nc = 1 - N;
}
double p2 = Math.Sqrt(-n * N);
double delta = Math.Sqrt(1 - k2 * sphi * sphi);
// Reduce A&S 17.7.15/16
// Mathematica eqns below
// N is protected in Mathematica, so use V
// V = (k2 - n)/(1 - n)
// Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[Sqrt[(1 - V)*(1 - k2/V)]/Sqrt[((1 - n)*(1 - k2/n))]]]
// Result: ((-1 + k2) n)/((-1 + n) (-k2 + n))
// Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[k2/(Sqrt[-n*(k2 - n)/(1 - n)]*Sqrt[(1 - n)*(1 - k2/n)])]]
// Result: k2/(k2 - n)
// Assuming[(k2 >= 0 && k2 <= 1) && n < 0, FullSimplify[Sqrt[1/((1 - n)*(1 - k2/n))]]]
// Result: Sqrt[n/((k2 - n) (-1 + n))]
double nResult = -(n / nc) * ((1 - k2) / (k2 - n)) * _EllintPi.Imp(k, N, Nc, rphi);
nResult += k2 / (k2 - n) * Math2.EllintF(k, rphi);
nResult += Math.Sqrt((n / nc) / (n - k2)) * Math.Atan((p2 / (2 * delta)) * Math.Sin(2 * rphi));
result += nResult;
return((phi < 0) ? -result : result);
}
double sinp = Math.Sin(rphi);
double cosp = Math.Cos(rphi);
double x = cosp * cosp;
double t = sinp * sinp;
double y = 1 - k * k * t;
double z = 1;
double p = (n * t < 0.5) ? 1 - n * t : x + nc * t;
result += sinp * (Math2.EllintRF(x, y, z) + n * t * Math2.EllintRJ(x, y, z, p) / 3);
return((phi < 0) ? -result : result);
}
// 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 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);
}