public class SinhTools extends java.lang.Object
Constructor and Description |
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SinhTools() |
Modifier and Type | Method and Description |
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static double |
sinhIntegral(double theta,
double b)
Computes an integral expression that occurs in the minimal solution of sinh-Gordon-type models.
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static double |
sinhIntegralCCPair(double theta,
Complex b)
Computes the product of two integral expression as in
sinhIntegral(double, double) but for a complex value \(B\)
and for its complex conjugate. |
static double |
sinhIntegralCCPairLimit(Complex b)
Computes the limit of the integral expression
sinhIntegralCCPair(double, org.apache.commons.math3.complex.Complex) as \(\theta \to \infty\). |
static double |
sinhIntegralLimit(double b)
Computes the limit of the integral expression
sinhIntegral(double, double) as \(\theta \to \infty\). |
public SinhTools()
public static double sinhIntegral(double theta, double b)
theta
- the parameter \( \theta \) in the integrandb
- the parameter \( B \) in the integrandpublic static double sinhIntegralCCPair(double theta, Complex b)
sinhIntegral(double, double)
but for a complex value \(B\)
and for its complex conjugate.
The expression is explicitly given by
\[ \exp 8\int_{0}^{\infty} \frac{dx}{x}\frac{ 2 \operatorname{Re} \Big(\sinh\Big(\frac{x B}{4}\Big)\sinh\Big(\frac{x}{2}(1-\frac{B}{2}) \Big) \Big)
\sinh\frac{x}{2}}{\sinh^2 x} \; \sin^2\Big( \frac{x\theta}{2\pi}\Big) \]
where \(\theta \in \mathbb{R} \) and \( B \in (0,2) + i \mathbb{R} \) are given as parameters.
The integral is numerically approximated using the composite Simpson's rule.theta
- the parameter \( \theta \) in the integrandb
- the complex parameter \( B \) in the integrandpublic static double sinhIntegralCCPairLimit(Complex b)
sinhIntegralCCPair(double, org.apache.commons.math3.complex.Complex)
as \(\theta \to \infty\).
By the Riemann-Lebesgue lemma, this limit is given by
\[ \exp 4\int_{0}^{\infty} \frac{dx}{x}\frac{ 2 \operatorname{Re} \Big(\sinh\Big(\frac{x B}{4}\Big)\sinh\Big(\frac{x}{2}(1-\frac{B}{2}) \Big) \Big)
\sinh\frac{x}{2}}{\sinh^2 x} \]
The integral is numerically approximated using the composite Simpson's rule.b
- the parameter \( B \) in the integrandpublic static double sinhIntegralLimit(double b)
sinhIntegral(double, double)
as \(\theta \to \infty\).
By the Riemann-Lebesgue lemma, this limit is given by
\[ \exp 4\int_{0}^{\infty} \frac{dx}{x}\frac{\sinh\Big(\frac{x B}{4}\Big)\sinh\Big(\frac{x}{2}(1-\frac{B}{2}) \Big)
\sinh\frac{x}{2}}{\sinh^2 x} \]
The integral is numerically approximated using the composite Simpson's rule.b
- the parameter \( B \) in the integrand