/// <summary>
/// Computes the complex Faddeeva function.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The complex value of w(z).</returns>
/// <remarks>
/// <para>The Faddeeva function w(z) is related to the error function with a complex argument.</para>
/// <img src="../images/FaddeevaErfcRelation.png" />
/// <para>It also has an integral representation.</para>
/// <img src="../images/FaddeevaIntegral.png" />
/// <para>For purely imaginary values, it reduces to the complementary error function (<see cref="AdvancedMath.Erfc"/>).
/// For purely real values, it reduces to Dawson's integral (<see cref="AdvancedMath.Dawson"/>).</para>
/// <para>It appears in the computation of the Voigt line profile function V(x;σ,γ).</para>
/// <img src="../images/Voigt.png" />
/// <para>Near the origin, w(z) ≈ 1. To accurately determine w(z) - 1 in this region, use the <see cref="Erf"/>
/// function. Away from the origin near the large negative imaginary axis, the magnitude w(z) increases rapidly and
/// may overflow.</para>
/// <para>The image below shows the complex Faddeeva function near the origin, using domain coloring.</para>
/// <img src="../images/ComplexFaddeevaPlot.png" />
/// </remarks>
/// <seealso cref="AdvancedComplexMath.Erf"/>
/// <seealso cref="AdvancedMath.Erf" />
/// <seealso cref="AdvancedMath.Erfc" />
/// <seealso cref="AdvancedMath.Dawson"/>
/// <seealso href="http://en.wikipedia.org/wiki/Voigt_profile" />
public static Complex Faddeeva(Complex z)
{
// use reflection formulae to ensure that we are in the first quadrant
if (z.Im < 0.0)
{
return(2.0 * ComplexMath.Exp(-z * z) - Faddeeva(-z));
}
if (z.Re < 0.0)
{
return(Faddeeva(-z.Conjugate).Conjugate);
}
double r = ComplexMath.Abs(z);
if (r < 2.0)
{
// use series for small z
return(ComplexMath.Exp(-z * z) * (1.0 - Erf_Series(-ComplexMath.I * z)));
//return (Faddeeva_Series(z));
}
else if ((z.Im < 0.1) && (z.Re < 30.0))
{
// this is a special, awkward region
// along the real axis, Re{w(x)} ~ e^{-x^2}; the Weideman algorthm doesen't compute this small number
// well and the Laplace continued fraction misses it entirely; therefore very close to the real axis
// we will use an analytic result on the real axis and Taylor expand to where we need to go.
// unfortunately the Taylor expansion converges poorly for large x, so we drop this work-arround near x~30,
// when this real part becomes too small to represent as a double anyway
double x = z.Re;
double y = z.Im;
return(Faddeeva_Taylor(new Complex(x, 0.0),
Math.Exp(-x * x) + 2.0 * AdvancedMath.Dawson(x) / Global.SqrtPI * ComplexMath.I,
new Complex(0.0, y)));
}
else if (r > 7.0)
{
// use Laplace continued fraction for large z
return(Faddeeva_ContinuedFraction(z));
}
else
{
// use Weideman algorithm for intermediate region
return(Faddeeva_Weideman(z));
}
}
