public void ComplexReimannZetaPrimesTest()
{
// pick high enough values so that p^-x == 1 within double precision before we reach the end of our list of primes
foreach (Complex z in TestUtilities.GenerateComplexValues(1.0, 100.0, 8))
{
Complex zz = z;
if (zz.Re < 0.0)
{
zz = -zz;
}
zz += 10.0;
Console.WriteLine(zz);
Complex f = 1.0;
for (int p = 2; p < 100; p++)
{
if (!AdvancedIntegerMath.IsPrime(p))
{
continue;
}
Complex t = Complex.One - ComplexMath.Pow(p, -zz);
if (t == Complex.One)
{
break;
}
f = f * t;
}
Assert.IsTrue(TestUtilities.IsNearlyEqual(1.0 / AdvancedComplexMath.RiemannZeta(zz), f));
}
}
public void ComplexRealRiemannZetaAgreement()
{
foreach (double x in TestUtilities.GenerateRealValues(1.0E-2, 1.0E2, 8))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedComplexMath.RiemannZeta(x), AdvancedMath.RiemannZeta(x)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedComplexMath.RiemannZeta(-x), AdvancedMath.RiemannZeta(-x)));
}
}
public static void ComputeSpecialFunctions()
{
// Compute the value x at which erf(x) is just 10^{-15} from 1.
double x = AdvancedMath.InverseErfc(1.0E-15);
// The Gamma function at 1/2 is sqrt(pi)
double y = AdvancedMath.Gamma(0.5);
// Compute a Coulomb Wave Function in the quantum tunneling region
SolutionPair s = AdvancedMath.Coulomb(2, 4.5, 3.0);
// Compute the Reimann Zeta function at a complex value
Complex z = AdvancedComplexMath.RiemannZeta(new Complex(0.75, 6.0));
// Compute the 100th central binomial coefficient
double c = AdvancedIntegerMath.BinomialCoefficient(100, 50);
}
public void ComplexRiemannZetaZeros()
{
// Zeros from http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros2
double[] zeros = new double[] {
14.134725141734693790,
21.022039638771554993,
25.010857580145688763,
30.424876125859513210
};
foreach (double zero in zeros)
{
double rho = FunctionMath.FindZero((double x) => {
Complex f = AdvancedComplexMath.RiemannZeta(new Complex(0.5, x));
return(f.Re + f.Im);
}, Math.Round(zero));
Assert.IsTrue(TestUtilities.IsNearlyEqual(rho, zero));
}
}
private void Render()
{
DeviceContext renderTarget = _deviceContext;
renderTarget.BeginDraw();
renderTarget.Clear(Color.FromKnown(Colors.Black, 1f));
for (double re = 0.5; re <= 2; re += 0.1d)
{
for (double im = -2; im <= 2; im += 0.1d)
{
Complex s = new Complex(re, im);
Complex z = AdvancedComplexMath.RiemannZeta(s);
PointF rs = TranslateToScreen(s.Real, s.Imaginary);
PointF rz = TranslateToScreen(z.Real, z.Imaginary);
renderTarget.DrawLine(_brush1, 1, rs, rz);
}
}
renderTarget.EndDraw();
_swapChain.Present(1, 0);
}
public static void Main(string[] args)
{
// calculate the factorials of 0 through 10
for (long counter = 0; counter <= 10; ++counter)
{
Console.WriteLine("{0}! = {1}", counter, Factorial(counter));
}
// Compute the value x at which erf(x) is just 10^{-15} from 1.
double x = AdvancedMath.InverseErfc(1.0E-15);
Console.WriteLine("\n{0}", x);
// The Gamma function at 1/2 is sqrt(pi)
double y = AdvancedMath.Gamma(0.5);
Console.WriteLine("\nThe Gamma function at 1/2 is sqrt(pi) = {0}", y);
// Compute a Coulomb Wave Function in the quantum tunneling region
SolutionPair s = AdvancedMath.Coulomb(2, 4.5, 3.0);
Console.WriteLine("\nCompute a Coulomb Wave Function in the quantum tunneling region = {0}", s);
// Compute the Reimann Zeta function at a complex value
Complex z = AdvancedComplexMath.RiemannZeta(new Complex(0.75, 6.0));
Console.WriteLine("\nCompute the Reimann Zeta function at a complex value = {0}", z);
// Compute the 100th central binomial coefficient
double c = AdvancedIntegerMath.BinomialCoefficient(5, 2);
Console.WriteLine("\n{0}", c);
Console.WriteLine("\nTecle qualquer tecla para finalizar...");
Console.ReadKey();
} // end Main