public void JacobiIntegrals()
{
foreach (double k in TestUtilities.GenerateRealValues(1.0E-1, 1.0, 4))
{
// DLMF 22.14.18 says
// \int_{0}^{K(k)} \ln(\dn(t, k)) \, dt = \frac{1}{2} K(k) \ln k'
double K = AdvancedMath.EllipticK(k);
double k1 = Math.Sqrt(1.0 - k * k);
double I1 = FunctionMath.Integrate(
x => Math.Log(AdvancedMath.JacobiDn(x, k)),
Interval.FromEndpoints(0.0, K)
);
Assert.IsTrue(TestUtilities.IsNearlyEqual(I1, K / 2.0 * Math.Log(k1)));
// If k is small, log(k1) is subject to cancellation error, so don't pick k too small.
// It also gives values for the analogous integrals with \sn and \cn,
// but their values involve K'(k), for which we don't have a method.
// Mathworld (http://mathworld.wolfram.com/CompleteEllipticIntegraloftheSecondKind.html) says
// \int_{0}^{K(k)} \dn^2(t, k) = E(k)
double I2 = FunctionMath.Integrate(
u => MoreMath.Sqr(AdvancedMath.JacobiDn(u, k)),
Interval.FromEndpoints(0.0, K)
);
Assert.IsTrue(TestUtilities.IsNearlyEqual(I2, AdvancedMath.EllipticE(k)));
}
}
public void EllipticLandenTransform()
{
foreach (double k in TestUtilities.GenerateUniformRealValues(0.0, 1.0, 8))
{
double kp = Math.Sqrt(1.0 - k * k);
double k1 = (1.0 - kp) / (1.0 + kp);
// For complete 1st kind
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticK(k),
(1.0 + k1) * AdvancedMath.EllipticK(k1)
));
// For complete 2nd kind
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticE(k) + kp * AdvancedMath.EllipticK(k),
(1.0 + kp) * AdvancedMath.EllipticE(k1)
));
/*
* foreach (double t in TestUtilities.GenerateUniformRealValues(0.0, Math.PI / 2.0, 4)) {
*
* double s = Math.Sin(t);
* double t1 = Math.Asin((1.0 + kp) * s * Math.Sqrt(1.0 - s * s) / Math.Sqrt(1.0 - k * k * s * s));
*
* // Since t1 can be > pi/2, we need to handle a larger range of angles before we can test the incomplete cases
*
*
* }
*
*/
}
}
public void EllipticECompleteAgreement()
{
foreach (double k in TestUtilities.GenerateRealValues(0.01, 1.0, 8))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticE(Math.PI / 2.0, k),
AdvancedMath.EllipticE(k)
));
}
}
public void EllipticLegendreRelation()
{
foreach (double k in TestUtilities.GenerateRealValues(0.01, 1.0, 8))
{
double kp = Math.Sqrt(1.0 - k * k);
double E = AdvancedMath.EllipticE(k);
double EP = AdvancedMath.EllipticE(kp);
double K = AdvancedMath.EllipticK(k);
double KP = AdvancedMath.EllipticK(kp);
Assert.IsTrue(TestUtilities.IsNearlyEqual(K * EP + KP * E, Math.PI / 2.0 + K * KP));
}
}
public void EllipticESPecialCases()
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.EllipticE(0.0), Math.PI / 2.0));
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.EllipticE(1.0), 1.0));
double g2 = Math.Pow(AdvancedMath.Gamma(1.0 / 4.0), 2.0);
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticE(1.0 / Math.Sqrt(2.0)),
Math.Pow(Math.PI, 3.0 / 2.0) / g2 + g2 / 8.0 / Math.Sqrt(Math.PI)
));
}
public void CompleteEllipticHypergeometricAgreement()
{
foreach (double k in TestUtilities.GenerateRealValues(1.0E-4, 1.0, 4))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticK(k),
Math.PI / 2.0 * AdvancedMath.Hypergeometric2F1(0.5, 0.5, 1.0, k * k)
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticE(k),
Math.PI / 2.0 * AdvancedMath.Hypergeometric2F1(0.5, -0.5, 1.0, k * k)
));
}
}
public void EllipticEDisplacement()
{
foreach (int m in TestUtilities.GenerateUniformIntegerValues(-100, 100, 4))
{
foreach (double phi in TestUtilities.GenerateUniformRealValues(-Math.PI / 2.0, +Math.PI / 2.0, 4))
{
foreach (double k in TestUtilities.GenerateRealValues(1.0E-4, 1.0, 4))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.EllipticE(m * Math.PI + phi, k),
2 * m * AdvancedMath.EllipticE(k) + AdvancedMath.EllipticE(phi, k)
));
}
}
}
}
public void EllipticEIntegration()
{
foreach (double k in TestUtilities.GenerateRealValues(1.0E-2, 1.0, 8))
{
// define the integrand
Func <double, double> f = delegate(double t) {
double z = k * Math.Sin(t);
return(Math.Sqrt(1.0 - z * z));
};
// test complete integral
double C = FunctionMath.Integrate(f, Interval.FromEndpoints(0.0, Math.PI / 2.0));
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.EllipticE(k), C));
// test incomplete integral
foreach (double phi in TestUtilities.GenerateUniformRealValues(0.0, Math.PI / 2.0, 8))
{
double I = FunctionMath.Integrate(f, Interval.FromEndpoints(0.0, phi));
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.EllipticE(phi, k), I));
}
}
}
public void HypergeometricBasicFunctions()
{
foreach (double x in xs)
{
// A&S 15.1.3
if (x < 1.0 && (x != 0.0))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(1.0, 1.0, 2.0, x), -Math.Log(1.0 - x) / x
));
}
// A&S 15.1.5
if (x != 0.0)
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(0.5, 1.0, 1.5, -x * x), Math.Atan(x) / x
));
}
// A&S 15.1.6
if ((-1.0 <= x) && (x <= 1.0) && (x != 0.0))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.Hypergeometric2F1(0.5, 0.5, 1.5, x * x), Math.Asin(x) / x
));
}
// Complete elliptic integrals
if ((0.0 <= x) && (x < 1.0))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
Math.PI / 2.0 * AdvancedMath.Hypergeometric2F1(0.5, 0.5, 1.0, x * x), AdvancedMath.EllipticK(x)
));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
Math.PI / 2.0 * AdvancedMath.Hypergeometric2F1(-0.5, 0.5, 1.0, x * x), AdvancedMath.EllipticE(x)
));
}
}
}
public void EllipticPiSpecialCharacteristic()
{
foreach (double k in TestUtilities.GenerateRealValues(1.0E-4, 1.0, 4))
{
// DLMF 19.6.1
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.EllipticPi(k * k, k), AdvancedMath.EllipticE(k) / (1.0 - k * k)));
// DLMF 19.6.2
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.EllipticPi(-k, k), Math.PI / 4.0 / (1.0 + k) + AdvancedMath.EllipticK(k) / 2.0));
// DLMF 19.6.3
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.EllipticPi(0.0, k), AdvancedMath.EllipticK(k)));
Assert.IsTrue(Double.IsPositiveInfinity(AdvancedMath.EllipticPi(1.0, k)));
}
}
public void EllipticEExtremeValues()
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.EllipticE(Double.Epsilon), Math.PI / 2.0));
Assert.IsTrue(Double.IsNaN(AdvancedMath.EllipticE(Double.NaN)));
}
