private static Complex LogGamma_Stirling(Complex z)
{
// work in the upper complex plane; i think this isn't actually necessary
//if (z.Im < 0.0) return (LogGamma_Stirling(z.Conjugate).Conjugate);
Complex f = (z - 0.5) * ComplexMath.Log(z) - z + 0.5 * Math.Log(Global.TwoPI);
// reduce f.Im modulo 2*PI
// result is cyclic in f.Im modulo 2*PI, but if f.Im starts off too big, the corrections
// applied below will be lost because they are being added to a big number
//f = new Complex(f.Re, AdvancedMath.Reduce(f.Im, 0.0));
// we should still do an optional reduction by 2 pi so that we get phase of Gamma right even
// for large z.
Complex zz = ComplexMath.Sqr(z);
Complex zp = z;
for (int i = 1; i < AdvancedIntegerMath.Bernoulli.Length; i++)
{
Complex f_old = f;
f += AdvancedIntegerMath.Bernoulli[i] / ((2 * i) * (2 * i - 1)) / zp;
if (f == f_old)
{
return(f);
}
zp *= zz;
}
throw new NonconvergenceException();
}
private static Complex LogGamma_Stirling(Complex z)
{
// work in the upper complex plane; i think this isn't actually necessary
if (z.Im < 0.0)
{
return(LogGamma_Stirling(z.Conjugate).Conjugate);
}
Complex f = (z - 0.5) * ComplexMath.Log(z) - z + Math.Log(Global.TwoPI) / 2.0;
// reduce f.Im modulo 2*PI
// result is cyclic in f.Im modulo 2*PI, but if f.Im starts off too big, the corrections
// applied below will be lost because they are being added to a big number
f = new Complex(f.Re, AdvancedMath.Reduce(f.Im, 0.0));
Complex zz = z * z;
Complex zp = z;
for (int i = 1; i < AdvancedIntegerMath.Bernoulli.Length; i++)
{
Complex f_old = f;
f += AdvancedIntegerMath.Bernoulli[i] / (2 * i) / (2 * i - 1) / zp;
if (f == f_old)
{
return(f);
}
zp *= zz;
}
throw new NonconvergenceException();
}
private static Complex DiLog_Series_1(Complex e)
{
Complex f = Math.PI * Math.PI / 6.0;
if (e == 0.0)
{
return(f);
}
Complex L = ComplexMath.Log(e);
Complex ek = 1.0;
for (int k = 1; k < Global.SeriesMax; k++)
{
Complex f_old = f;
ek *= e;
Complex df = ek * (L - 1.0 / k) / k;
f += df;
if (f == f_old)
{
return(f);
}
}
throw new NonconvergenceException();
}
public static Complex LogGamma(Complex z)
{
Complex t = z + LanczosGP;
return(
Math.Log(Global.SqrtTwoPI) +
(z - 0.5) * ComplexMath.Log(t) - t +
ComplexMath.Log(Sum(z))
);
}
/// <summary>
/// Computes the complex dilogarithm function, also called Spence's function.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The value Li<sub>2</sub>(z).</returns>
/// <remarks>
/// <para>This function is the analytic continuation of the dilogarithm function (<see cref="AdvancedMath.DiLog"/>) into the complex plane.</para>
/// <para>The image below shows the complex dilogarithm function near the origin, using domain coloring.</para>
/// <img src="../images/ComplexDiLogPlot.png" />
/// </remarks>
/// <seealso cref="AdvancedMath.DiLog"/>
/// <seealso href="http://mathworld.wolfram.com/Dilogarithm.html" />
public static Complex DiLog(Complex z)
{
Complex f;
double a0 = ComplexMath.Abs(z);
if (a0 > 1.0)
{
// outside the unit disk, reflect into the unit disk
Complex ln = ComplexMath.Log(-z);
f = -Math.PI * Math.PI / 6.0 - ln * ln / 2.0 - DiLog(1.0 / z);
}
else
{
// inside the unit disk...
if (a0 < 0.75)
{
// close to 0, use the expansion about zero
f = DiLog_Series_0(z);
}
else
{
// we are in the annulus near the edge of the unit disk
if (z.Re < 0.0)
{
// reflect negative into positive half-disk
// this avoids problems with the log expansion near -1
f = DiLog(z * z) / 2.0 - DiLog(-z);
}
else
{
// figure out whether we are close to 1
Complex e = 1.0 - z;
if (ComplexMath.Abs(e) < 0.5)
{
// close to 1, use the expansion about 1
f = DiLog_Series_1(e);
}
else
{
// otherwise, use the log expansion, which is good
// near the unit circle but not too close to 1 or -1
f = DiLog_Log_Series(z);
}
}
}
}
if ((z.Re > 1.0) && (Math.Sign(f.Im) != Math.Sign(z.Im)))
{
f = f.Conjugate;
}
return(f);
}
public void TestLog()
{
Complex arg;
Complex result;
arg = new Complex(1 / 2.0, 1 / 3.0);
Assert.AreEqual(0.6009252125773315488532035, arg.GetModulus(), 1e-15);
result = ComplexMath.Log(arg);
Assert.AreEqual(-0.5092847904972866327857336, result.Re, 1e-15);
Assert.AreEqual(0.5880026035475675512456111, result.Im, 1e-15);
}
private static Complex LogGammaReflectionTerm(Complex z)
{
// s will overflow for large z.Im, but we are taking its log, so we should
// re-formulate this
Complex s = new Complex(MoreMath.SinPi(z.Re) * Math.Cosh(Math.PI * z.Im), MoreMath.CosPi(z.Re) * Math.Sinh(Math.PI * z.Im));
Complex logs = ComplexMath.Log(s);
double m1 = Math.Log(MoreMath.Hypot(MoreMath.SinPi(z.Re), Math.Sinh(Math.PI * z.Im)));
double t = Math.Abs(Math.PI * z.Im);
double c1 = Math.Exp(-2.0 * t);
double c2 = MoreMath.SinPi(z.Re) / Math.Sinh(t);
double m2 = t + MoreMath.LogOnePlus(-c1) + 0.5 * MoreMath.LogOnePlus(c2 * c2) - Math.Log(2.0);
return(Math.Log(Math.PI) - logs);
}
public void Log()
{
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.Log(cd1);
Assert.AreEqual(cdt.Real, 0.900, TOLERENCE);
Assert.AreEqual(cdt.Imag, -1.107, TOLERENCE);
cdt = ComplexMath.Log(cd2);
Assert.AreEqual(cdt.Real, 0.788, TOLERENCE);
Assert.AreEqual(cdt.Imag, -1.571, TOLERENCE);
cdt = ComplexMath.Log(cd3);
Assert.AreEqual(cdt.Real, 0.0953, TOLERENCE);
Assert.AreEqual(cdt.Imag, 0, TOLERENCE);
cdt = ComplexMath.Log(cd4);
Assert.AreEqual(cdt.Real, 0.900, TOLERENCE);
Assert.AreEqual(cdt.Imag, 2.034, TOLERENCE);
ComplexFloat cft = ComplexMath.Log(cf1);
Assert.AreEqual(cft.Real, 0.900, TOLERENCE);
Assert.AreEqual(cft.Imag, -1.107, TOLERENCE);
cft = ComplexMath.Log(cf2);
Assert.AreEqual(cft.Real, 0.788, TOLERENCE);
Assert.AreEqual(cft.Imag, -1.571, TOLERENCE);
cft = ComplexMath.Log(cf3);
Assert.AreEqual(cft.Real, 0.095, TOLERENCE);
Assert.AreEqual(cft.Imag, 0, TOLERENCE);
cft = ComplexMath.Log(cf4);
Assert.AreEqual(cft.Real, 0.900, TOLERENCE);
Assert.AreEqual(cft.Imag, 2.034, TOLERENCE);
}
/// <summary>
/// Compute the complex log Gamma function.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The principal complex value y for which exp(y) = Γ(z).</returns>
/// <seealso cref="AdvancedMath.LogGamma" />
/// <seealso href="http://mathworld.wolfram.com/LogGammaFunction.html"/>
public static Complex LogGamma(Complex z)
{
if (z.Im == 0.0 && z.Re < 0.0)
{
// Handle the pure negative case explicitly.
double re = Math.Log(Math.PI / Math.Abs(MoreMath.SinPi(z.Re))) - AdvancedMath.LogGamma(1.0 - z.Re);
double im = Math.PI * Math.Floor(z.Re);
return(new Complex(re, im));
}
else if (z.Re > 16.0 || Math.Abs(z.Im) > 16.0)
{
// According to https://dlmf.nist.gov/5.11, the Stirling asymptoic series is valid everywhere
// except on the negative real axis. So at first I tried to use it for |z.Im| > 0, |z| > 16. But in practice,
// it exhibits false convergence close to the negative real axis, e.g. z = -16 + i. So I have
// moved to requiring |z| large and reasonably far from the negative real axis.
return(Stirling.LogGamma(z));
}
else if (z.Re >= 0.125)
{
return(Lanczos.LogGamma(z));
}
else
{
// For the remaining z < 0, we need to use the reflection formula.
// For large z.Im, SinPi(z) \propto e^{\pi |z.Im|} overflows even though its log does not.
// It's possible to do some algebra to get around that problem, but it's not necessary
// because for z.Im that big we would have used the Stirling series.
Complex f = ComplexMath.Log(Math.PI / ComplexMath.SinPi(z));
Complex g = Lanczos.LogGamma(1.0 - z);
// The reflection formula doesn't stay on the principal branch, so we need to add a multiple of 2 \pi i
// to fix it up. See Hare, "Computing the Principal Branch of Log Gamma" for how to do this.
// https://pdfs.semanticscholar.org/1c9d/8865836a312836500126cb47c3cbbed3043e.pdf
Complex h = new Complex(0.0, 2.0 * Math.PI * Math.Floor(0.5 * (z.Re + 0.5)));
if (z.Im < 0.0)
{
h = -h;
}
return(f - g + h);
}
}
private static Complex DiLog_Log_Series(Complex z)
{
Complex ln = ComplexMath.Log(z);
Complex ln2 = ln * ln;
Complex f = Math.PI * Math.PI / 6.0 + ln * (1.0 - ComplexMath.Log(-ln)) - ln2 / 4.0;
Complex p = ln;
for (int k = 1; k < AdvancedIntegerMath.Bernoulli.Length; k++)
{
Complex f_old = f;
p *= ln2 / (2 * k + 1) / (2 * k);
f += (-AdvancedIntegerMath.Bernoulli[k] / (2 * k)) * p;
if (f == f_old)
{
return(f);
}
}
throw new NonconvergenceException();
}
public void EvaluateGradient(double[] X, double[] P, double[][] DY)
{
double x = X[0];
if (_useFrequencyInsteadOmega)
{
x *= (2 * Math.PI);
}
DY[0][0] = 1;
DY[1][0] = 0;
Complex OneByDenom = 1 / ComplexMath.Pow(1 + ComplexMath.Pow(Complex.I * x * P[2], P[3]), P[4]);
DY[0][1] = OneByDenom.Re;
DY[1][1] = -OneByDenom.Im;
Complex IXP2 = Complex.I * x * P[2];
Complex IXP2PowP3 = ComplexMath.Pow(IXP2, P[3]);
Complex der2 = OneByDenom * -P[1] * P[2] * P[4] * IXP2PowP3 / (P[2] * (1 + IXP2PowP3));
DY[0][2] = der2.Re;
DY[1][2] = -der2.Im;
Complex der3 = OneByDenom * -P[1] * P[4] * IXP2PowP3 *ComplexMath.Log(IXP2) / (1 + IXP2PowP3);
DY[0][3] = der3.Re;
DY[1][3] = -der3.Im;
Complex der4 = OneByDenom * -P[1] * ComplexMath.Log(1 + IXP2PowP3);
DY[0][4] = der4.Re;
DY[1][4] = -der4.Im;
if (_useFlowTerm)
{
DY[0][5] = 0;
DY[1][5] = _isDielectricData ? 1 / (x * 8.854187817e-12) : 1 / x;
}
}
/// <summary>
/// Computes the entire complex exponential integral.
/// </summary>
/// <param name="z">The complex argument.</param>
/// <returns>The value of Ein(z).</returns>
/// <remarks>
/// <para>The entire exponential integral function can be defined by an integral or an equivalent series.</para>
/// <img src="..\images\EinIntegralSeries.png" />
/// <para>Both Ei(x) and E<sub>1</sub>(z) can be obtained from Ein(z).</para>
/// <img src="..\images\E1EiEinRelation.png" />
/// <para>Unlike either Ei(x) or E<sub>1</sub>(z), Ein(z) is entire, that is, it has no poles or cuts anywhere
/// in the complex plane.</para>
/// </remarks>
/// <seealso cref="AdvancedMath.IntegralEi(double)"/>
/// <seealso cref="AdvancedMath.IntegralE(int, double)"/>
public static Complex Ein(Complex z)
{
if (z.Re < -40.0)
{
// For sufficiently negative z, use the asymptotic expansion for Ei
return(AdvancedMath.EulerGamma + ComplexMath.Log(-z) - IntegralEi_AsymptoticSeries(-z));
}
else
{
// Ideally, we would like to use the series within some simple radius of the origin and the continued fraction
// outside it. That works okay for the right half-plane, but for z.Re < 0 we have to contend with the fact that the
// continued fraction fails on or near the negative real axis. That makes us use the series in an oddly shaped
// region that extends much further from the origin than we would like into the left half-plane. It would be
// great if we could find an alternative approach in that region.
if (IsEinSeriesPrefered(z))
{
return(Ein_Series(z));
}
else
{
return(AdvancedMath.EulerGamma + ComplexMath.Log(z) + IntegeralE1_ContinuedFraction(z));
}
}
}
public static Complex LogGamma(Complex z)
{
return((z - 0.5) * ComplexMath.Log(z) - z + halfLogTwoPi + Sum(z));
}
public static Complex Psi(Complex z)
{
Complex t = z + LanczosGP;
return(ComplexMath.Log(t) - LanczosG / t + LogSumPrime(z));
}
