//
// Series form for BesselY as z -> 0,
// see: http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/06/01/04/01/01/0003/
// This series is only useful when the second term is small compared to the first
// otherwise we get catestrophic cancellation errors.
//
// Approximating tgamma(v) by v^v, and assuming |tgamma(-z)| < eps we end up requiring:
// eps/2 * v^v(x/2)^-v > (x/2)^v or Math.Log(eps/2) > v Math.Log((x/2)^2/v)
//
/// <summary>
/// Series form for Y{v}(x) as z -> 0 and v is not an integer
/// <para>Y{v}(z) = (-((z/2)^v Cos(πv) Γ(-v))/π)) * 0F1(; 1+v; -z^2/4) - ((z/2)^-v Γ(v))/π) * 0F1(; 1-v; -z^2/4)</para>
/// </summary>
/// <param name="v"></param>
/// <param name="x"></param>
/// <returns></returns>
/// <seealso href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselY/26/01/02/0001/"/>
public static DoubleX Y_SmallArgSeries(double v, double x)
{
Debug.Assert(v >= 0 && x >= 0);
Debug.Assert( !Math2.IsInteger(v), "v cannot be an integer");
DoubleX powDivGamma;
double lnp = v * Math.Log(x / 2);
if (v > DoubleLimits.MaxGamma || lnp < DoubleLimits.MinLogValue || lnp > DoubleLimits.MaxLogValue) {
powDivGamma = DoubleX.Exp(lnp - Math2.Lgamma(v));
} else {
DoubleX p = Math.Pow(x / 2, v);
DoubleX gam = Math2.Tgamma(v);
powDivGamma = p / gam;
}
// Series 1: -((z/2)^-v Γ(v))/π) * 0F1(; 1-v; -z^2/4)
double s1 = HypergeometricSeries.Sum0F1(1 - v, -x * x / 4);
DoubleX result1 = -(s1/Math.PI)/ powDivGamma;
// Series2: -((z/2)^v Cos(πv) Γ(-v))/π)) * 0F1(; 1+v; -z^2/4)
// using the reflection formula: Γ(-v) = -π/(Γ(v) * v * Sin(π * v))
// prefix = (z/2)^v * Cos(π * v)/(Γ(v) * v * Sin(π * v))
// prefix = (z/2)^v * Cot(π * v) / (Γ(v) * v)
DoubleX result2 = DoubleX.Zero;
double cot = Math2.CotPI(v);
if (cot != 0) {
double s2 = HypergeometricSeries.Sum0F1(v + 1, -x * x / 4);
result2 = powDivGamma * cot * s2 / v; // Do this all in DoubleX
}
return (result1 - result2);
}
/// <summary>
/// Returns the Digamma function
/// <para>ψ(x) = d/dx(ln(Γ(x))) = Γ'(x)/Γ(x)</para>
/// </summary>
/// <param name="x">Digamma function argument</param>
public static double Digamma(double x)
{
if (double.IsNaN(x))
{
Policies.ReportDomainError("Digamma(x: {0}): NaN not allowed", x);
return(double.NaN);
}
if (double.IsInfinity(x))
{
if (x < 0)
{
Policies.ReportDomainError("Digamma(x: {0}): Requires x != -Infinity", x);
return(double.NaN);
}
return(double.PositiveInfinity);
}
if (x <= 0)
{
if (IsInteger(x))
{
Policies.ReportPoleError("Digamma(x: {0}) Requires x is not an integer when x <= 0", x);
return(double.NaN);
}
// use the following equations to reflect
// ψ(1-x) - ψ(x) = π * cot(π*x)
// ψ(-x) = 1/x + ψ(x) + π * cot(π*x)
if (x < -1)
{
return(Digamma(1 - x) - Math.PI * Math2.CotPI(x));
}
// The following is the forward recurrence:
// -(1/x + 1/(x+1)) + Digamma(x + 2);
// with a little less cancellation error near Digamma root: x = -0.50408...
return(_Digamma.Rational_1_2(x + 2) - (2 * x + 1) / (x * (x + 1)));
}
// If we're above the lower-limit for the asymptotic expansion then use it:
if (x >= 10)
{
const double p0 = 0.083333333333333333333333333333333333333333333333333;
const double p1 = -0.0083333333333333333333333333333333333333333333333333;
const double p2 = 0.003968253968253968253968253968253968253968253968254;
const double p3 = -0.0041666666666666666666666666666666666666666666666667;
const double p4 = 0.0075757575757575757575757575757575757575757575757576;
const double p5 = -0.021092796092796092796092796092796092796092796092796;
const double p6 = 0.083333333333333333333333333333333333333333333333333;
const double p7 = -0.44325980392156862745098039215686274509803921568627;
double xm1 = x - 1;
double z = 1 / (xm1 * xm1);
double P = p0 + z * (p1 + z * (p2 + z * (p3 + z * (p4 + z * (p5 + z * (p6 + z * p7))))));
return(Math.Log(xm1) + 1.0 / (2 * xm1) - z * P);
}
double result = 0;
//
// If x > 2 reduce to the interval [1,2]:
//
while (x > 2)
{
x -= 1;
result += 1 / x;
}
//
// If x < 1 use recurrence to shift to > 1:
//
if (x < 1)
{
result = -1 / x;
x += 1;
}
result += _Digamma.Rational_1_2(x);
return(result);
}