public void DoubleFactorialSpecialCases()
{
Assert.IsTrue(AdvancedIntegerMath.DoubleFactorial(0) == 1);
Assert.IsTrue(AdvancedIntegerMath.DoubleFactorial(1) == 1);
Assert.IsTrue(AdvancedIntegerMath.DoubleFactorial(2) == 2);
Assert.IsTrue(AdvancedIntegerMath.DoubleFactorial(3) == 3);
Assert.IsTrue(AdvancedIntegerMath.DoubleFactorial(4) == 8);
Assert.IsTrue(AdvancedIntegerMath.DoubleFactorial(5) == 15);
}
/// <inheritdoc />
public override double CentralMoment(int r)
{
if (r < 0)
{
throw new ArgumentOutOfRangeException(nameof(r));
}
else if (r == 0)
{
return(1.0);
}
else if ((r % 2) == 0)
{
// (r-1)!! \sigma^r
return(AdvancedIntegerMath.DoubleFactorial(r - 1) * MoreMath.Pow(sigma, r));
}
else
{
return(0.0);
}
}
/// <inheritdoc />
public override double Cumulant(int r)
{
if (r < 0)
{
throw new ArgumentOutOfRangeException("r");
}
else if (r == 0)
{
return(0.0);
}
else if (r == 1)
{
return(Mean);
}
else
{
// K_{r+1} = \frac{(2 r)!}{2^r r! \mu^{2r+1} \lambda^r} = \frac{(2r - 1)!! \mu^{2r+1}}{\lambda^r}
return(AdvancedIntegerMath.DoubleFactorial(2 * r - 3) * MoreMath.Pow(mu, 2 * r - 1) / MoreMath.Pow(lambda, r - 1));
}
}
/// <inheritdoc />
public override double Cumulant(int r)
{
if (r < 0)
{
throw new ArgumentOutOfRangeException(nameof(r));
}
else if (r == 0)
{
return(0.0);
}
else if (r == 1)
{
return(Mean);
}
else
{
// Mathworld (http://mathworld.wolfram.com/InverseGaussianDistribution.html) says:
// K_{r+1} = \frac{(2 r)!}{2^r r! \mu^{2r+1} \lambda^r}
// = \frac{(2r - 1)!! \mu^{2r+1}}{\lambda^r}
return(AdvancedIntegerMath.DoubleFactorial(2 * r - 3) * MoreMath.Pow(mu * mu / lambda, r - 1) * mu);
}
}
public override double MomentAboutMean(int n)
{
if (n < 0)
{
throw new ArgumentOutOfRangeException("n");
}
else if (n == 0)
{
return(1.0);
}
else if (n % 2 != 0)
{
return(0.0);
}
else
{
// Using
// \int_0^D x^{2m} e^{-x^2 / 2} = 2^{m-1/2} \left[ \Gamma(m+1/2) - \Gamma(m+1/2, D^2 / 2) \right]
// and \Gamma(m + 1/2) = \frac{(2m-1)!!}{2^m} \sqrt{\pi} and normalization above, you get
// C_{2m} = \frac{(2m-1)!!}{\erf(D/\sqrt{2}) \left[ 1 - \frac{\Gamma(m+1/2, D^2 / 2)}{\Gamma(m + 1/2)} \right]
// which reduced to standard normal (2m-1)!! in D \rightarrow \infty limit as expected.
return(AdvancedIntegerMath.DoubleFactorial(n - 1) * (1.0 - AdvancedMath.RightRegularizedGamma(n / 2 + 0.5, D * D / 2.0)) / ED);
}
}
public void SphericalBesselMomentIntegral()
{
// \int_0^{\infty} dx x^n j_{n+1}(x) = 1/(2n+1)!!
// which follows from derivative formula
// (https://math.stackexchange.com/questions/805379/integral-involving-the-spherical-bessel-function-of-the-first-kind-int-0)
foreach (int n in TestUtilities.GenerateIntegerValues(4, 100, 4))
{
IntegrationResult M = FunctionMath.Integrate(x => MoreMath.Pow(x, -n) * AdvancedMath.SphericalBesselJ(n + 1, x), 0.0, Double.PositiveInfinity);
Assert.IsTrue(M.Estimate.ConfidenceInterval(0.95).ClosedContains(1.0 / AdvancedIntegerMath.DoubleFactorial(2 * n + 1)));
}
}