/// <summary>
/// Computes the Riemann zeta function for complex values.
/// </summary>
/// <param name="z">The argument.</param>
/// <returns>The value of ζ(z).</returns>
/// <remarks>
/// <para>As the imaginary part of the argument increases, the computation of the zeta function becomes slower and more difficult.
/// The computation time is approximately proportional to the imaginary part of z. The result also slowly looses accuracy for arguments with
/// very large imaginary parts; for arguments with z.Im of order 10^d, approximately the last d digits of the result are suspect.</para>
/// <para>The image below shows the complex Γ function near the origin using domain coloring. You can see the first non-trivial
/// zeros at (1/2, ±14.13...) as well as the trivial zeros along the negative real axis.</para>
/// <img src="../images/ComplexRiemannZetaPlot.png" />
/// </remarks>
public static Complex RiemannZeta(Complex z)
{
// Use conjugation and reflection symmetry to move to the first quadrant.
if (z.Im < 0.0)
{
return(RiemannZeta(z.Conjugate).Conjugate);
}
if (z.Re < 0.0)
{
Complex zp = Complex.One - z;
return(2.0 * ComplexMath.Pow(Global.TwoPI, -zp) * ComplexMath.Cos(Global.HalfPI * zp) * AdvancedComplexMath.Gamma(zp) * RiemannZeta(zp));
}
// Close to pole, use Laurent series.
Complex zm1 = z - Complex.One;
if (ComplexMath.Abs(zm1) < 0.50)
{
return(RiemannZeta_LaurentSeries(zm1));
}
// Fall back to Euler-Maclaurin summation.
int n = RiemannZeta_EulerMaclaurin_N(z.Re, z.Im);
return(RiemannZeta_EulerMaclaurin(z, n));
}
public void Cos()
{
Complex cd1 = new Complex(1.1, -2.2);
Complex cd2 = new Complex(0, -2.2);
Complex cd3 = new Complex(1.1, 0);
Complex cd4 = new Complex(-1.1, 2.2);
ComplexFloat cf1 = new ComplexFloat(1.1f, -2.2f);
ComplexFloat cf2 = new ComplexFloat(0, -2.2f);
ComplexFloat cf3 = new ComplexFloat(1.1f, 0);
ComplexFloat cf4 = new ComplexFloat(-1.1f, 2.2f);
Complex cdt = ComplexMath.Cos(cd1);
Assert.AreEqual(cdt.Real, 2.072, TOLERENCE);
Assert.AreEqual(cdt.Imag, 3.972, TOLERENCE);
cdt = ComplexMath.Cos(cd2);
Assert.AreEqual(cdt.Real, 4.568, TOLERENCE);
Assert.AreEqual(cdt.Imag, 0, TOLERENCE);
cdt = ComplexMath.Cos(cd3);
Assert.AreEqual(cdt.Real, 0.454, TOLERENCE);
Assert.AreEqual(cdt.Imag, 0, TOLERENCE);
cdt = ComplexMath.Cos(cd4);
Assert.AreEqual(cdt.Real, 2.072, TOLERENCE);
Assert.AreEqual(cdt.Imag, 3.972, TOLERENCE);
ComplexFloat cft = ComplexMath.Cos(cf1);
Assert.AreEqual(cft.Real, 2.072, TOLERENCE);
Assert.AreEqual(cft.Imag, 3.972, TOLERENCE);
cft = ComplexMath.Cos(cf2);
Assert.AreEqual(cft.Real, 4.568, TOLERENCE);
Assert.AreEqual(cft.Imag, 0, TOLERENCE);
cft = ComplexMath.Cos(cf3);
Assert.AreEqual(cft.Real, 0.454, TOLERENCE);
Assert.AreEqual(cft.Imag, 0, TOLERENCE);
cft = ComplexMath.Cos(cf4);
Assert.AreEqual(cft.Real, 2.072, TOLERENCE);
Assert.AreEqual(cft.Imag, 3.972, TOLERENCE);
}
public static Complex AiryAi_Asymptotic(Complex z)
{
Debug.Assert(ComplexMath.Abs(z) >= 9.0);
if (z.Re >= 0.0)
{
Complex xi = 2.0 / 3.0 * ComplexMath.Pow(z, 3.0 / 2.0);
Airy_Asymptotic_Subseries(xi, out Complex u0, out Complex v0, out Complex u1, out Complex v1);
Complex e = ComplexMath.Exp(xi);
Complex q = ComplexMath.Pow(z, 1.0 / 4.0);
return(0.5 / Global.SqrtPI / q / e * u1);
}
else
{
z = -z;
Complex xi = 2.0 / 3.0 * ComplexMath.Pow(z, 3.0 / 2.0);
Airy_Asymptotic_Subseries(xi, out Complex u0, out Complex v0, out Complex u1, out Complex v1);
Complex c = ComplexMath.Cos(xi);
Complex s = ComplexMath.Sin(xi);
throw new NotImplementedException();
}
}
