// series for L=0 for both F and G
// this has the same convergence properties as the L != 0 series for F above
private static void Coulomb_Zero_Series(double eta, double rho, out double F, out double FP, out double G, out double GP)
{
if (rho == 0.0)
{
double C = CoulombFactorZero(eta);
F = 0.0;
FP = C;
G = 1.0 / C;
GP = Double.NegativeInfinity;
return;
}
double eta_rho = eta * rho;
double rho_2 = rho * rho;
double u0 = 0.0;
double u1 = rho;
double u = u0 + u1;
double up = u1;
double v0 = 1.0;
double v1 = 0.0;
double v = v0 + v1;
for (int n = 2; n <= Global.SeriesMax; n++)
{
double u2 = (2.0 * eta_rho * u1 - rho_2 * u0) / (n * (n - 1));
double v2 = (2.0 * eta_rho * v1 - rho_2 * v0 - 2.0 * eta * (2 * n - 1) * u2) / (n * (n - 1));
double u_old = u;
u += u2;
up += n * u2;
double v_old = v;
v += v2;
if ((u == u_old) && (v == v_old))
{
double C = CoulombFactorZero(eta);
F = C * u;
FP = C * up / rho;
double r = AdvancedComplexMath.Psi(new Complex(1.0, eta)).Re + 2.0 * AdvancedMath.EulerGamma - 1.0;
G = (v + 2.0 * eta * u * (Math.Log(2.0 * rho) + r)) / C;
GP = (FP * G - 1.0) / F;
return;
}
u0 = u1; u1 = u2; v0 = v1; v1 = v2;
}
throw new NonconvergenceException();
}
// asymptotic region
private static void Coulomb_Asymptotic(double L, double eta, double rho, out double F, out double G)
{
// compute phase
// reducing the eta = 0 and eta != 0 parts seperately preserves accuracy for large rho and small eta
double t0 = Reduce(rho, -L / 4.0);
double t1 = Reduce(AdvancedComplexMath.LogGamma(new Complex(L + 1.0, eta)).Im - eta * Math.Log(2.0 * rho), 0.0);
double t = t0 + t1;
double s = Math.Sin(t);
double c = Math.Cos(t);
// determine the weights of sin and cos
double f0 = 1.0;
double g0 = 0.0;
double f = f0;
double g = g0;
for (int k = 0; true; k++)
{
// compute the next contributions to f and g
double q = 2 * (k + 1) * rho;
double a = (2 * k + 1) * eta / q;
double b = ((L * (L + 1) - k * (k + 1)) + eta * eta) / q;
double f1 = a * f0 - b * g0;
double g1 = a * g0 + b * f0;
// add them
double f_old = f;
f += f1;
double g_old = g;
g += g1;
if ((f == f_old) && (g == g_old))
{
break;
}
// check for non-convergence
if (k > Global.SeriesMax)
{
throw new NonconvergenceException();
}
// prepare for the next iteration
f0 = f1;
g0 = g1;
}
F = g * c + f * s;
G = f * c - g * s;
}
/// <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));
}
// asymptotic region
private static SolutionPair Coulomb_Asymptotic(double L, double eta, double rho)
{
// Abromowitz & Stegun 14.5 describes this asymptotic expansion
/*
* double F, G;
* Coulomb_Asymptotic(L, eta, rho, out F, out G);
* return (new SolutionPair(F, 0.0, G, 0.0));
*/
/*
* // Old reduction algorithm
* double t0 = Reduce(rho, -L / 4.0);
* double t1 = Reduce(AdvancedComplexMath.LogGamma(new Complex(L + 1.0, eta)).Im - eta * Math.Log(2.0 * rho), 0.0);
* double t = t0 + t1;
* double s = Math.Sin(t);
* double c = Math.Cos(t);
*/
/*
* // New reduction algorithm
* double t = AdvancedComplexMath.LogGamma(new Complex(L + 1.0, eta)).Im - eta * Math.Log(2.0 * rho) + rho;
* double s = MoreMath.Sin(t);
* double c = MoreMath.Cos(t);
*
* int r = -((int) L) % 4;
* if (r < 0) r += 4;
*
* switch (r) {
* case 0:
* // no change
* break;
* case 1:
* Global.Swap(ref s, ref c);
* c = -c;
* break;
* case 2:
* s = -s;
* c = -c;
* break;
* case 3:
* Global.Swap(ref s, ref c);
* s = -s;
* break;
* default:
* throw new InvalidOperationException();
* };
*/
long t00; double t01;
RangeReduction.ReduceByPiHalves(rho, out t00, out t01);
long t10; double t11;
RangeReduction.ReduceByPiHalves(AdvancedComplexMath.LogGamma(new Complex(L + 1.0, eta)).Im - eta * Math.Log(2.0 * rho), out t10, out t11);
long t20; double t1;
RangeReduction.ReduceByOnes(t01 + t11, out t20, out t1);
long t0 = t00 + t10 + t20 - (long)L;
double s = RangeReduction.Sin(t0, t1);
double c = RangeReduction.Cos(t0, t1);
/*
* long tt0; double tt1;
* RangeReduction.ReduceByPiHalves(rho - eta * Math.Log(2.0 * rho) + AdvancedComplexMath.LogGamma(new Complex(L + 1.0, eta)).Im, out tt0, out tt1);
*
* tt0 -= (long) L;
*
* double s = RangeReduction.Sin(tt0, tt1);
* double c = RangeReduction.Cos(tt0, tt1);
*/
/*
* // compute phase
* // reducing the eta = 0 and eta != 0 parts seperately preserves accuracy for large rho and small eta
* double t0 = Reduce(rho, -L / 4.0);
* double t1 = Reduce(AdvancedComplexMath.LogGamma(new Complex(L + 1.0, eta)).Im - eta * Math.Log(2.0 * rho), 0.0);
* double t = t0 + t1;
* double s = Math.Sin(t);
* double c = Math.Cos(t);
*/
// determine the weights of sin and cos
double f0 = 1.0;
double g0 = 0.0;
double fp0 = 0.0;
double gp0 = 1.0 - eta / rho;
double f = f0;
double g = g0;
double fp = fp0;
double gp = gp0;
for (int k = 0; k < Global.SeriesMax; k++)
{
// Remember the old values for comparison
double f_old = f;
double g_old = g;
double fp_old = fp;
double gp_old = gp;
// Compute the next corrections
double q = 2 * (k + 1) * rho;
double a = (2 * k + 1) * eta / q;
double b = ((L * (L + 1) - k * (k + 1)) + eta * eta) / q;
double f1 = a * f0 - b * g0;
double g1 = a * g0 + b * f0;
double fp1 = a * fp0 - b * gp0 - f1 / rho;
double gp1 = a * gp0 + b * fp0 - g1 / rho;
// Apply them
f += f1;
g += g1;
fp += fp1;
gp += gp1;
// Test for convergence
if ((f == f_old) && (g == g_old) && (fp == fp_old))
{
return(new SolutionPair(
g * c + f * s,
gp * c + fp * s,
f * c - g * s,
fp * c - gp * s)
);
}
// Prepare for the next iteration
f0 = f1;
g0 = g1;
fp0 = fp1;
gp0 = gp1;
}
throw new NonconvergenceException();
}
// asymptotic region
private static SolutionPair Coulomb_Asymptotic(double L, double eta, double rho)
{
// Abromowitz & Stegun 14.5 describes this asymptotic expansion
RangeReduction.ReduceByPiHalves(rho, out long t00, out double t01);
RangeReduction.ReduceByPiHalves(AdvancedComplexMath.LogGamma(new Complex(L + 1.0, eta)).Im - eta * Math.Log(2.0 * rho), out long t10, out double t11);
RangeReduction.ReduceByOnes(t01 + t11, out long t20, out double t1);
long t0 = t00 + t10 + t20 - (long)L;
double s = RangeReduction.Sin(t0, t1);
double c = RangeReduction.Cos(t0, t1);
// determine the weights of sin and cos
double f0 = 1.0;
double g0 = 0.0;
double fp0 = 0.0;
double gp0 = 1.0 - eta / rho;
double f = f0;
double g = g0;
double fp = fp0;
double gp = gp0;
for (int k = 0; k < Global.SeriesMax; k++)
{
// Remember the old values for comparison
double f_old = f;
double g_old = g;
double fp_old = fp;
//double gp_old = gp;
// Compute the next corrections
double q = 2 * (k + 1) * rho;
double a = (2 * k + 1) * eta / q;
double b = ((L * (L + 1) - k * (k + 1)) + eta * eta) / q;
double f1 = a * f0 - b * g0;
double g1 = a * g0 + b * f0;
double fp1 = a * fp0 - b * gp0 - f1 / rho;
double gp1 = a * gp0 + b * fp0 - g1 / rho;
// Apply them
f += f1;
g += g1;
fp += fp1;
gp += gp1;
// Test for convergence
if ((f == f_old) && (g == g_old) && (fp == fp_old))
{
return(new SolutionPair(
g * c + f * s,
gp * c + fp * s,
f * c - g * s,
fp * c - gp * s)
);
}
// Prepare for the next iteration
f0 = f1;
g0 = g1;
fp0 = fp1;
gp0 = gp1;
}
throw new NonconvergenceException();
}