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);
}