/// <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 proprotional 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 Evaluate(double[] X, double[] P, double[] Y)
{
double x = X[0];
if (_useFrequencyInsteadOmega)
{
x *= (2 * Math.PI);
}
Complex result = P[0] + P[1] / ComplexMath.Pow(1 + ComplexMath.Pow(Complex.I * x * P[2], P[3]), P[4]);
Y[0] = result.Re;
if (this._useFlowTerm)
{
if (this._isDielectricData)
{
Y[1] = -result.Im + P[5] / (x * 8.854187817e-12);
}
else
{
Y[1] = -result.Im + P[5] / (x);
}
}
else
{
Y[1] = -result.Im;
}
}
private static void CalcClx(string param, StreamWriter writer)
{
var x = new Complex(param);
var ans = ComplexMath.Pow(x, x) - x * x + 1;
writer.WriteLine(ans.ToString());
}
public void Evaluate(double[] X, double[] P, double[] Y)
{
double x = X[0];
if (_useFrequencyInsteadOmega)
{
x *= (2 * Math.PI);
}
Complex result = 1 / ComplexMath.Pow(1 + ComplexMath.Pow(Complex.I * x * P[2], P[3]), P[4]);
result = P[1] + (P[0] - P[1]) * result;
if (_useFlowTerm)
{
result = 1 / ((1 / result) - Complex.I * P[5] / x);
}
if (_logarithmizeResults)
{
Y[0] = Math.Log10(result.Re);
Y[1] = Math.Log10(result.Im);
}
else
{
Y[0] = result.Re;
Y[1] = result.Im;
}
}
public void Evaluate(double[] X, double[] P, double[] Y)
{
double x = X[0];
if (_useFrequencyInsteadOmega)
{
x *= (2 * Math.PI);
}
// Model this first as compliance
Complex result = 1 / ComplexMath.Pow(1 + ComplexMath.Pow(Complex.I * x * P[2], P[3]), P[4]);
result = (1 / P[1]) + ((1 / P[0]) - (1 / P[1])) * result;
if (_useFlowTerm)
{
result -= Complex.I * P[5] / x;
}
result = 1 / result; // but now convert to modulus
if (_logarithmizeResults)
{
Y[0] = Math.Log10(result.Re);
Y[1] = Math.Log10(result.Im);
}
else
{
Y[0] = result.Re;
Y[1] = result.Im;
}
}
public static Complex Gamma(Complex z)
{
Complex t = z + LanczosGP;
return(
Global.SqrtTwoPI *
ComplexMath.Pow(t / Math.E, z - 0.5) * LanczosExpG *
Sum(z)
);
}
// Euler-Maclaurin summation approximates a sum by an integral.
// \sum_{k=a}^{b} f(k) = \int_a^b \! f(x) \, dx - B_1 \left[ f(b) + f(a) \right] +
// \sum_{j=0}^{\infty} \frac{B_{2j}}{(2j)!} \left[ f^{(2j-1)}(b) - f^{(2j-1)}(a) \right]
private static Complex RiemannZeta_EulerMaclaurin(Complex z, int n)
{
Complex f = 0.0;
// Sum terms up to (n-1)^{-z}
for (int k = 1; k < n; k++)
{
f += ComplexMath.Pow(k, -z);
}
// Integral term
Complex f_old = f;
Complex zm1 = z - Complex.One;
f += ComplexMath.Pow(n, -zm1) / zm1;
if (f == f_old)
{
return(f);
}
// B_1 term
f_old = f;
Complex df = ComplexMath.Pow(n, -z) / 2.0;
f += df;
if (f == f_old)
{
return(f);
}
// B_2 term
df *= z / n;
f += AdvancedIntegerMath.BernoulliNumber(2) * df;
if (f == f_old)
{
return(f);
}
// Higher Bernoulli terms
for (int k = 4; k < 2 * AdvancedIntegerMath.Bernoulli.Length; k += 2)
{
f_old = f;
df *= (z + (k - 2)) / n * (z + (k - 3)) / n / k / (k - 1);
f += AdvancedIntegerMath.Bernoulli[k / 2] * df;
if (f == f_old)
{
return(f);
}
}
throw new NonconvergenceException();
}
public void Evaluate(double[] X, double[] P, double[] Y)
{
double x = X[0];
if (_useFrequencyInsteadOmega)
{
x *= (2 * Math.PI);
}
Complex result = P[0];
int i, j;
for (i = 0, j = 1; i < _numberOfTerms; ++i, j += 4)
{
result += P[j] / ComplexMath.Pow(1 + ComplexMath.Pow(Complex.I * x * P[1 + j], P[2 + j]), P[3 + j]);
}
// note: because it is a susceptiblity, the imaginary part is still negative
if (_useFlowTerm)
{
if (_isDielectricData)
{
result.Im -= P[j] / (x * 8.854187817e-12);
}
else if (_invertViscosity)
{
result.Im -= P[j] / (x);
}
else
{
result.Im -= 1 / (P[j] * x);
}
}
if (_invertResult)
{
result = 1 / result; // if we invert, i.e. we calculate the modulus, the imaginary part is now positive
}
else
{
result.Im = -result.Im; // else if we don't invert, i.e. we calculate susceptibility, we negate the imaginary part to make it positive
}
if (_logarithmizeResults)
{
result.Re = Math.Log10(result.Re);
result.Im = Math.Log10(result.Im);
}
Y[0] = result.Re;
Y[1] = result.Im;
}
public static Complex Riemann_Euler(Complex z)
{
Complex f = 1.0;
for (int k = 0; k < primes.Length; k++)
{
Complex f_old = f;
Complex fk = 1.0 - ComplexMath.Pow(primes[k], -z);
f = f * fk;
if (f == f_old)
{
return(1.0 / f);
}
}
throw new NonconvergenceException();
}
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();
}
}
public void EvaluateGradient(double[] X, double[] P, double[][] DY)
{
double x = X[0];
if (_useFrequencyInsteadOmega)
{
x *= (2 * Math.PI);
}
DY[0][0] = 1;
DY[1][0] = 0;
Complex OneByDenom = 1 / ComplexMath.Pow(1 + ComplexMath.Pow(Complex.I * x * P[2], P[3]), P[4]);
DY[0][1] = OneByDenom.Re;
DY[1][1] = -OneByDenom.Im;
Complex IXP2 = Complex.I * x * P[2];
Complex IXP2PowP3 = ComplexMath.Pow(IXP2, P[3]);
Complex der2 = OneByDenom * -P[1] * P[2] * P[4] * IXP2PowP3 / (P[2] * (1 + IXP2PowP3));
DY[0][2] = der2.Re;
DY[1][2] = -der2.Im;
Complex der3 = OneByDenom * -P[1] * P[4] * IXP2PowP3 *ComplexMath.Log(IXP2) / (1 + IXP2PowP3);
DY[0][3] = der3.Re;
DY[1][3] = -der3.Im;
Complex der4 = OneByDenom * -P[1] * ComplexMath.Log(1 + IXP2PowP3);
DY[0][4] = der4.Re;
DY[1][4] = -der4.Im;
if (_useFlowTerm)
{
DY[0][5] = 0;
DY[1][5] = _isDielectricData ? 1 / (x * 8.854187817e-12) : 1 / x;
}
}
