public override Complex Value(
double radius_fm
)
{
double X = DebyeMass_MeV * radius_fm / Constants.HbarC_MeV_fm;
if (radius_fm == 0)
{
return(new Complex(0, 0));
}
else if (X <= 10)
{
return(new Complex(-(AlphaEff / radius_fm + SigmaOverDebyeMass_fm) * Math.Exp(-X),
-AlphaOverDebyeMass_fm
* (1.0 - 0.5 * (AdvancedMath.IntegralEi(X) * Math.Exp(-X) * (1.0 / X + 1.0)
+ AdvancedMath.IntegralE(1, X) * Math.Exp(X) * (1.0 / X - 1.0)))));
}
else
{
double X2 = X * X;
return(new Complex(-(AlphaEff / radius_fm + SigmaOverDebyeMass_fm) * Math.Exp(-X),
-AlphaOverDebyeMass_fm
* (1.0 - (((40.0 / X2 + 1.0) * 18.0 / X2 + 1.0) * 4.0 / X2 + 1.0) * 2.0 / X2)));
}
}
public void IntegralE0()
{
// A & S 5.1.24
foreach (double x in TestUtilities.GenerateRealValues(1.0E-4, 1.0E3, 8))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.IntegralE(0, x), Math.Exp(-x) / x));
}
}
public void IntegralEAtZero()
{
// A & S 5.1.23
foreach (int n in TestUtilities.GenerateIntegerValues(2, 100, 8))
{
Assert.IsTrue(AdvancedMath.IntegralE(n, 0.0) == 1.0 / (n - 1.0));
}
}
public void IntegralE1Integral()
{
// A & S 5.1.33
Assert.IsTrue(TestUtilities.IsNearlyEqual(
FunctionMath.Integrate(t => MoreMath.Sqr(AdvancedMath.IntegralE(1, t)), 0.0, Double.PositiveInfinity),
2.0 * Math.Log(2.0)
));
}
public void IntegralEiE1Integrals()
{
// Here are a couple unusual integrals involving both Ei and E1
// https://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn3p191_A1b.pdf
Assert.IsTrue(TestUtilities.IsNearlyEqual(
FunctionMath.Integrate(x => Math.Exp(-x) * AdvancedMath.IntegralE(1, x) * AdvancedMath.IntegralEi(x), 0.0, Double.PositiveInfinity),
-Math.PI * Math.PI / 12.0
));
}
public void ComplexEinE1Agreement()
{
foreach (double x in TestUtilities.GenerateRealValues(1.0E-3, 1.0E3, 8))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedComplexMath.Ein(x),
AdvancedMath.IntegralE(1, x) + AdvancedMath.EulerGamma + Math.Log(x)
));
}
}
public void IntegralESpecialCases()
{
foreach (int n in TestUtilities.GenerateIntegerValues(1, 100, 4))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(AdvancedMath.IntegralE(n, 0.0), 1.0 / (n - 1)));
Assert.IsTrue(AdvancedMath.IntegralE(n, Double.MaxValue) == 0.0);
Assert.IsTrue(AdvancedMath.IntegralE(n, Double.PositiveInfinity) == 0.0);
Assert.IsTrue(Double.IsNaN(AdvancedMath.IntegralE(n, Double.NaN)));
}
}
public void IntegralE1Inequality()
{
// A & S 5.1.20
// Choose maximum x to keep e^x from overflowing.
foreach (double x in TestUtilities.GenerateRealValues(1.0E-4, Math.Log(Double.MaxValue / 10.0), 8))
{
double lower = Math.Log(1.0 + 2.0 / x) / 2.0;
double upper = Math.Log(1.0 + 1.0 / x);
double value = Math.Exp(x) * AdvancedMath.IntegralE(1, x);
Assert.IsTrue((lower < value) && (value < upper));
}
}
public void IntegralERecurrence()
{
// A & S 5.1.14
foreach (int n in TestUtilities.GenerateIntegerValues(1, 100, 8))
{
foreach (double x in TestUtilities.GenerateRealValues(1.0E-4, 1.0E4, 24))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
n * AdvancedMath.IntegralE(n + 1, x) + x * AdvancedMath.IntegralE(n, x),
Math.Exp(-x)
));
}
}
}
public void IntegralEDefinition()
{
// A & S 5.1.4
foreach (int n in TestUtilities.GenerateIntegerValues(1, 100, 4))
{
foreach (double x in TestUtilities.GenerateRealValues(1.0E-2, 10.0, 8))
{
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.IntegralE(n, x),
FunctionMath.Integrate(t => Math.Exp(-x * t) / MoreMath.Pow(t, n), 1.0, Double.PositiveInfinity)
));
}
}
}
public void IntegralEInequality()
{
// A & S 5.1.19
foreach (int n in TestUtilities.GenerateIntegerValues(1, 100, 8))
{
// Choose maximum x to keep e^x from overflowing.
foreach (double x in TestUtilities.GenerateRealValues(1.0E-4, Math.Log(Double.MaxValue / 10.0), 8))
{
double lower = 1.0 / (x + n);
double upper = 1.0 / (x + (n - 1));
double value = Math.Exp(x) * AdvancedMath.IntegralE(n, x);
Assert.IsTrue((lower < value) && (value <= upper));
}
}
}
public void ComplexEinAgreement () {
foreach (double x in TestUtilities.GenerateRealValues(1.0E-2, 1.0E4, 16)) {
Console.WriteLine("Ei {0} {1} {2}", x, AdvancedMath.IntegralEi(x), AdvancedMath.EulerGamma + Math.Log(x) - AdvancedComplexMath.Ein(-x));
if (x < Math.Log(Double.MaxValue / 10.0)) {
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.IntegralEi(x),
AdvancedMath.EulerGamma + Math.Log(x) - AdvancedComplexMath.Ein(-x)
));
}
Console.WriteLine("E1 {0} {1} {2}", x, AdvancedMath.IntegralE(1, x) + AdvancedMath.EulerGamma + Math.Log(x), AdvancedComplexMath.Ein(x));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
AdvancedMath.IntegralE(1, x) + AdvancedMath.EulerGamma + Math.Log(x),
AdvancedComplexMath.Ein(x)
));
}
}
