public void TestExp()
{
Complex result;
result = ComplexMath.Exp(new Complex(0.5, 0.5));
Assert.AreEqual(1.446889036584169158051583, result.Re, 1e-15);
Assert.AreEqual(0.7904390832136149118432626, result.Im, 1e-15);
}
/// <summary>
/// Computes the complex Gamma function.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The complex value of Γ(z).</returns>
/// <remarks>
/// <para>The image below shows the complex Γ function near the origin using domain coloring.</para>
/// <img src="../images/ComplexGammaPlot.png" />
/// </remarks>
/// <seealso cref="AdvancedMath.Gamma(double)"/>
/// <seealso href="http://en.wikipedia.org/wiki/Gamma_function" />
/// <seealso href="http://mathworld.wolfram.com/GammaFunction.html" />
public static Complex Gamma(Complex z)
{
if (z.Re < 0.5)
{
// 1-z form
return(Math.PI / Gamma(1.0 - z) / ComplexMath.Sin(Math.PI * z));
// -z form
//return (-Math.PI / Gamma(-z) / z / ComplexMath.Sin(Math.PI * z));
}
return(ComplexMath.Exp(LogGamma(z)));
}
private static Complex IntegralEi_AsymptoticSeries(Complex z)
{
Complex dy = 1.0;
Complex y = dy;
for (int k = 1; k < Global.SeriesMax; k++)
{
Complex y_old = y;
dy = dy * k / z;
y = y_old + dy;
if (y == y_old)
{
return(ComplexMath.Exp(z) / z * y);
}
}
throw new NonconvergenceException();
}
/// <summary>
/// Computes the complex Gamma function.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The complex value of Γ(z).</returns>
/// <remarks>
/// <para>The image below shows the complex Γ function near the origin using domain coloring.</para>
/// <img src="../images/ComplexGammaPlot.png" />
/// </remarks>
/// <seealso cref="AdvancedMath.Gamma(double)"/>
/// <seealso href="http://en.wikipedia.org/wiki/Gamma_function" />
/// <seealso href="http://mathworld.wolfram.com/GammaFunction.html" />
public static Complex Gamma(Complex z)
{
if (z.Re < 0.5)
{
// 1-z form
return(Math.PI / Gamma(1.0 - z) / ComplexMath.SinPi(z));
}
else if (ComplexMath.Abs(z) < 16.0)
{
return(Lanczos.Gamma(z));
}
else
{
// Add flag to do z-reduction
return(ComplexMath.Exp(LogGamma_Stirling(z)));
}
}
/// <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));
}
}
public void Exp()
{
Complex cd1 = new Complex(1.1, -2.2);
Complex cd2 = new Complex(0, -2.2);
Complex cd3 = new Complex(1.1, 0);
Complex cd4 = new Complex(-1.1, 2.2);
ComplexFloat cf1 = new ComplexFloat(1.1f, -2.2f);
ComplexFloat cf2 = new ComplexFloat(0, -2.2f);
ComplexFloat cf3 = new ComplexFloat(1.1f, 0);
ComplexFloat cf4 = new ComplexFloat(-1.1f, 2.2f);
Complex cdt = ComplexMath.Exp(cd1);
Assert.AreEqual(cdt.Real, -1.768, TOLERENCE);
Assert.AreEqual(cdt.Imag, -2.429, TOLERENCE);
cdt = ComplexMath.Exp(cd2);
Assert.AreEqual(cdt.Real, -0.589, TOLERENCE);
Assert.AreEqual(cdt.Imag, -0.808, TOLERENCE);
cdt = ComplexMath.Exp(cd3);
Assert.AreEqual(cdt.Real, 3.004, TOLERENCE);
Assert.AreEqual(cdt.Imag, 0, TOLERENCE);
cdt = ComplexMath.Exp(cd4);
Assert.AreEqual(cdt.Real, -0.196, TOLERENCE);
Assert.AreEqual(cdt.Imag, 0.269, TOLERENCE);
ComplexFloat cft = ComplexMath.Exp(cf1);
Assert.AreEqual(cft.Real, -1.768, TOLERENCE);
Assert.AreEqual(cft.Imag, -2.429, TOLERENCE);
cft = ComplexMath.Exp(cf2);
Assert.AreEqual(cft.Real, -0.589, TOLERENCE);
Assert.AreEqual(cft.Imag, -0.808, TOLERENCE);
cft = ComplexMath.Exp(cf3);
Assert.AreEqual(cft.Real, 3.004, TOLERENCE);
Assert.AreEqual(cft.Imag, 0, TOLERENCE);
cft = ComplexMath.Exp(cf4);
Assert.AreEqual(cft.Real, -0.196, TOLERENCE);
Assert.AreEqual(cft.Imag, 0.269, TOLERENCE);
}
// This continued fraction is valid for |z| >> 1, except close to the negative real axis.
private static Complex IntegeralE1_ContinuedFraction(Complex z)
{
int a = 1; // a_1
Complex b = z + 1.0; // b_1
Complex D = 1.0 / b; // D_1 = 1 / b_1 (denominator of a_1 / b_1)
Complex Df = a / b; // Df_1 = f_1 - f_0 = a_1 / b_1
Complex f = 0.0 + Df; // f_1 = f_0 + Df_1 = b_0 + a_1 / b_1 (here b_0 = 0)
for (int k = 1; k < Global.SeriesMax; k++)
{
Complex f_old = f;
a = -k * k;
b += 2.0;
D = 1.0 / (b + a * D);
Df = (b * D - 1.0) * Df;
f += Df;
if (f == f_old)
{
return(ComplexMath.Exp(-z) * f);
}
}
throw new NonconvergenceException();
}
public static Complex AiryAi_Asymptotic(Complex z)
{
Debug.Assert(ComplexMath.Abs(z) >= 9.0);
if (z.Re >= 0.0)
{
Complex xi = 2.0 / 3.0 * ComplexMath.Pow(z, 3.0 / 2.0);
Airy_Asymptotic_Subseries(xi, out Complex u0, out Complex v0, out Complex u1, out Complex v1);
Complex e = ComplexMath.Exp(xi);
Complex q = ComplexMath.Pow(z, 1.0 / 4.0);
return(0.5 / Global.SqrtPI / q / e * u1);
}
else
{
z = -z;
Complex xi = 2.0 / 3.0 * ComplexMath.Pow(z, 3.0 / 2.0);
Airy_Asymptotic_Subseries(xi, out Complex u0, out Complex v0, out Complex u1, out Complex v1);
Complex c = ComplexMath.Cos(xi);
Complex s = ComplexMath.Sin(xi);
throw new NotImplementedException();
}
}
/// <summary>
/// Computes the complex error function.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The value of erf(z).</returns>
/// <remarks>
/// <para>This function is the analytic continuation of the error function (<see cref="AdvancedMath.Erf"/>) to the complex plane.</para>
/// <para>The image below shows the complex error function near the origin, using domain coloring.</para>
/// <img src="../images/ComplexErfPlot.png" />
/// <para>The complex error function is entire: it has no poles, cuts, or discontinuities anywhere in the complex plane.</para>
/// <para>For pure imaginary arguments, erf(z) reduces to the Dawson integral (<see cref="AdvancedMath.Dawson"/>).</para>
/// <para>Away from the origin near the real axis, the real part of erf(z) quickly approaches ±1. To accurately determine
/// the small difference erf(z) ∓ 1 in this region, use the <see cref="Faddeeva"/> function. Away from the origin near
/// the imaginary axis, the magnitude of erf(z) increases very quickly. Although erf(z) may overflow in this region, you
/// can still accurately determine the value of the product erf(z) exp(z<sup>2</sup>) using the <see cref="Faddeeva"/>
/// function.</para>
/// </remarks>
/// <seealso cref="AdvancedMath.Erf"/>
/// <seealso cref="AdvancedMath.Dawson"/>
/// <seealso cref="AdvancedComplexMath.Faddeeva"/>
public static Complex Erf(Complex z)
{
double r = ComplexMath.Abs(z);
if (r < 4.0)
{
// near the origin, use the series
return(Erf_Series(z));
}
else
{
// otherwise, just compute from Faddeva
if (z.Re < 0.0)
{
// since Fadddeeva blows up for negative z.Re, use erf(z) = -erf(-z)
return(ComplexMath.Exp(-z * z) * Faddeeva(-ComplexMath.I * z) - 1.0);
}
else
{
return(1.0 - ComplexMath.Exp(-z * z) * Faddeeva(ComplexMath.I * z));
}
// we don't do this near the origin beause we would loose accuracy in the very small real parts there by subtracting from 1
}
}
private static Complex ComplexTestFunction(
double x
)
{
return(ComplexMath.Exp(new Complex(-x, -x)));
}