/// <summary>
/// Returns the value of the Airy Ai function at <paramref name="x"/>
/// </summary>
/// <param name="x">The function argument</param>
/// <returns></returns>
public static double AiryAi(double x)
{
if (double.IsNaN(x))
{
Policies.ReportDomainError("AiryAi(x: {0}): NaN not allowed", x);
return(double.NaN);
}
if (double.IsInfinity(x))
{
return(0);
}
double absX = Math.Abs(x);
if (x >= -3 && x <= 2)
{
// Use small series. See:
// http://functions.wolfram.com/Bessel-TypeFunctions/AiryAi/06/01/02/01/0004/
// http://dlmf.nist.gov/9.4
double z = x * x * x / 9;
double s1 = Constants.AiryAi0 * HypergeometricSeries.Sum0F1(2 / 3.0, z);
double s2 = x * Constants.AiryAiPrime0 * HypergeometricSeries.Sum0F1(4 / 3.0, z);
return(s1 + s2);
}
const double v = 1.0 / 3;
double zeta = 2 * absX * Math.Sqrt(absX) / 3;
if (x < 0)
{
if (x < -32)
{
return(AiryAsym.AiBiNeg(x).Ai);
}
// The following is
//double j1 = Math2.BesselJ(v, zeta);
//double j2 = Math2.BesselJ(-v, zeta);
//double ai = Math.Sqrt(-x) * (j1 + j2) / 3;
var(J, Y) = _Bessel.JY(v, zeta, true, true);
double s = 0.5 * (J - ((double)Y) / Constants.Sqrt3);
double ai = Math.Sqrt(-x) * s;
return(ai);
}
else
{
// Use the K relationship: http://dlmf.nist.gov/9.6
if (x >= 16)
{
return(AiryAsym.Ai(x));
}
double ai = Math2.BesselK(v, zeta) * Math.Sqrt(x / 3) / Math.PI;
return(ai);
}
}
/// <summary>
/// Returns the value of first derivative to the Airy Bi function at <paramref name="x"/>
/// </summary>
/// <param name="x">AiryBiPrime function argument</param>
/// <returns></returns>
public static double AiryBiPrime(double x)
{
if (double.IsNaN(x))
{
Policies.ReportDomainError("AiryBiPrime(x: {0}): NaN not allowed", x);
return(double.NaN);
}
if (double.IsInfinity(x))
{
if (x < 0)
{
return(0);
}
return(double.PositiveInfinity);
}
double absX = Math.Abs(x);
if (x >= -3 && x <= 3)
{
// Use small series. See:
// http://functions.wolfram.com/Bessel-TypeFunctions/AiryBiPrime/26/01/01/
// http://dlmf.nist.gov/9.4
double z = x * x * x / 9;
double s1 = x * x * (Constants.AiryBi0 / 2) * HypergeometricSeries.Sum0F1(5 / 3.0, z);
double s2 = Constants.AiryBiPrime0 * HypergeometricSeries.Sum0F1(1 / 3.0, z);
return(s1 + s2);
}
const double v = 2.0 / 3;
double zeta = 2 * absX * Math.Sqrt(absX) / 3;
if (x < 0)
{
if (x < -32)
{
return(AiryAsym.AiBiPrimeNeg(x).Bi);
}
// The following is
//double j1 = Math2.BesselJ(v, zeta);
//double j2 = Math2.BesselJ(-v, zeta);
//double bip = -x * (j1 + j2) / Constants.Sqrt3;
//return bip;
var(J, Y) = _Bessel.JY(v, zeta, true, true);
double s = 0.5 * (J / Constants.Sqrt3 - ((double)Y));
double bip = -x * s;
return(bip);
}
else
{
if (x >= 16)
{
return(AiryAsym.BiPrime(x));
}
// The following is
//double j1 = Math2.BesselI(v, zeta);
//double j2 = Math2.BesselI(-v, zeta);
//double bip = (x / Constants.Sqrt3) * (j1 + j2);
var(I, K) = _Bessel.IK(v, zeta, true, true);
double s = (2 / Constants.Sqrt3 * I + K / Math.PI);
var bip = x * s;
return(bip);
}
}
/// <summary>
/// Returns the value of the first derivative to the Airy Ai function at <paramref name="x"/>
/// </summary>
/// <param name="x">The function argument</param>
/// <returns></returns>
public static double AiryAiPrime(double x)
{
if (double.IsNaN(x))
{
Policies.ReportDomainError("AiryAiPrime(x: {0}): NaN not allowed", x);
return(double.NaN);
}
if (double.IsInfinity(x))
{
if (x > 0)
{
return(0);
}
Policies.ReportDomainError("AiryAiPrime(x: {0}): Requires x != -Infinity", x);
return(double.NaN);
}
double absX = Math.Abs(x);
if (x >= -3 && x <= 2)
{
// Use small series. See:
// http://functions.wolfram.com/Bessel-TypeFunctions/AiryAiPrime/26/01/01/
// http://dlmf.nist.gov/9.4
double z = x * x * x / 9;
double s1 = x * x * (Constants.AiryAi0 / 2) * HypergeometricSeries.Sum0F1(5 / 3.0, z);
double s2 = Constants.AiryAiPrime0 * HypergeometricSeries.Sum0F1(1 / 3.0, z);
return(s1 + s2);
}
const double v = 2.0 / 3;
double zeta = 2 * absX * Math.Sqrt(absX) / 3;
if (x < 0)
{
if (x < -32)
{
return(AiryAsym.AiBiPrimeNeg(x).Ai);
}
// The following is
//double j1 = Math2.BesselJ(v, zeta);
//double j2 = Math2.BesselJ(-v, zeta);
//double aip = -x * (j1 - j2) / 3;
var(J, Y) = _Bessel.JY(v, zeta, true, true);
double s = 0.5 * (J + ((double)Y) / Constants.Sqrt3);
double aip = -x * s;
return(aip);
}
else
{
if (x >= 16)
{
return(AiryAsym.AiPrime(x));
}
double aip = -Math2.BesselK(v, zeta) * x / (Constants.Sqrt3 * Math.PI);
return(aip);
}
}