// 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();
}
private static double PolyLog_BernoulliSum(int n, double w)
{
double s = 0.0;
for (int k = n; k > 1; k--)
{
double ds = (1.0 - MoreMath.Pow(2.0, 1 - k)) * Math.Abs(AdvancedIntegerMath.BernoulliNumber(k)) *
MoreMath.Pow(Global.TwoPI, k) / AdvancedIntegerMath.Factorial(k) *
MoreMath.Pow(w, n - k) / AdvancedIntegerMath.Factorial(n - k);
s -= ds;
}
s -= MoreMath.Pow(w, n) / AdvancedIntegerMath.Factorial(n);
return(s);
}