/// <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)));
}
static void Main(string[] args)
{
// usage
ComplexNumber i = 2;
ComplexNumber j = new ComplexNumber(1, -2);
Console.WriteLine(i * j);
Console.WriteLine(ComplexMath.Power(j, 2));
Console.WriteLine((double)ComplexMath.Sin(i) + " vs " + Math.Sin(2));
Console.WriteLine(ComplexMath.Power(j, 0) == 1.0);
}
public void Sin()
{
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.Sin(cd1);
Assert.AreEqual(cdt.Real, 4.071, TOLERENCE);
Assert.AreEqual(cdt.Imag, -2.022, TOLERENCE);
cdt = ComplexMath.Sin(cd2);
Assert.AreEqual(cdt.Real, 0, TOLERENCE);
Assert.AreEqual(cdt.Imag, -4.457, TOLERENCE);
cdt = ComplexMath.Sin(cd3);
Assert.AreEqual(cdt.Real, 0.891, TOLERENCE);
Assert.AreEqual(cdt.Imag, 0, TOLERENCE);
cdt = ComplexMath.Sin(cd4);
Assert.AreEqual(cdt.Real, -4.071, TOLERENCE);
Assert.AreEqual(cdt.Imag, 2.022, TOLERENCE);
ComplexFloat cft = ComplexMath.Sin(cf1);
Assert.AreEqual(cft.Real, 4.071, TOLERENCE);
Assert.AreEqual(cft.Imag, -2.022, TOLERENCE);
cft = ComplexMath.Sin(cf2);
Assert.AreEqual(cft.Real, 0, TOLERENCE);
Assert.AreEqual(cft.Imag, -4.457, TOLERENCE);
cft = ComplexMath.Sin(cf3);
Assert.AreEqual(cft.Real, 0.891, TOLERENCE);
Assert.AreEqual(cft.Imag, 0, TOLERENCE);
cft = ComplexMath.Sin(cf4);
Assert.AreEqual(cft.Real, -4.071, TOLERENCE);
Assert.AreEqual(cft.Imag, 2.022, TOLERENCE);
}
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();
}
}