Class LegendreEllipticIntegral
The elliptic integrals are related to Jacobi elliptic functions.
Beware that when computing elliptic integrals in the complex plane, many issues arise due to branch cuts. See the user guide for a thorough explanation.
There are different conventions to interpret the arguments of Legendre elliptic integrals. In mathematical texts, these conventions show up using the separator between arguments. So for example for the incomplete integral of the first kind F we have:
- F(φ, k): the first argument φ is an angle and the second argument k is the elliptic modulus: this is the trigonometric form of the integral
- F(φ; m): the first argument φ is an angle and the second argument m=k² is the parameter: this is also a trigonometric form of the integral
- F(x|m): the first argument x=sin(φ) is not an angle anymore and the second argument m=k² is the parameter: this is the Legendre form
- F(φ\α): the first argument φ is an angle and the second argument α is the modular angle
As we have no separator in a method call, we have to adopt one convention and stick to it. In Hipparchus, we adopted the Legendre form (i.e. F(x|m), with x=sin(φ) and m=k². These conventions are consistent with Wolfram Alpha functions EllipticF, EllipticE, ElliptiPI…
- Since:
- 2.0
- See Also:
-
Method Summary
Modifier and TypeMethodDescriptionstatic double
bigD
(double m) Get the complete elliptic integral D(m) = [K(m) - E(m)]/m.static double
bigD
(double phi, double m) Get the incomplete elliptic integral D(φ, m) = [F(φ, m) - E(φ, m)]/m.static Complex
Get the complete elliptic integral D(m) = [K(m) - E(m)]/m.static Complex
Get the incomplete elliptic integral D(φ, m) = [F(φ, m) - E(φ, m)]/m.static <T extends CalculusFieldElement<T>>
FieldComplex<T>bigD
(FieldComplex<T> m) Get the complete elliptic integral D(m) = [K(m) - E(m)]/m.static <T extends CalculusFieldElement<T>>
FieldComplex<T>bigD
(FieldComplex<T> phi, FieldComplex<T> m) Get the incomplete elliptic integral D(φ, m) = [F(φ, m) - E(φ, m)]/m.static <T extends CalculusFieldElement<T>>
TbigD
(T m) Get the complete elliptic integral D(m) = [K(m) - E(m)]/m.static <T extends CalculusFieldElement<T>>
TbigD
(T phi, T m) Get the incomplete elliptic integral D(φ, m) = [F(φ, m) - E(φ, m)]/m.static double
bigE
(double m) Get the complete elliptic integral of the second kind E(m).static double
bigE
(double phi, double m) Get the incomplete elliptic integral of the second kind E(φ, m).static Complex
Get the complete elliptic integral of the second kind E(m).static Complex
Get the incomplete elliptic integral of the second kind E(φ, m).static Complex
bigE
(Complex phi, Complex m, ComplexUnivariateIntegrator integrator, int maxEval) Get the incomplete elliptic integral of the second kind E(φ, m) using numerical integration.static <T extends CalculusFieldElement<T>>
FieldComplex<T>bigE
(FieldComplex<T> m) Get the complete elliptic integral of the second kind E(m).static <T extends CalculusFieldElement<T>>
FieldComplex<T>bigE
(FieldComplex<T> phi, FieldComplex<T> m) Get the incomplete elliptic integral of the second kind E(φ, m).static <T extends CalculusFieldElement<T>>
FieldComplex<T>bigE
(FieldComplex<T> phi, FieldComplex<T> m, FieldComplexUnivariateIntegrator<T> integrator, int maxEval) Get the incomplete elliptic integral of the second kind E(φ, m).static <T extends CalculusFieldElement<T>>
TbigE
(T m) Get the complete elliptic integral of the second kind E(m).static <T extends CalculusFieldElement<T>>
TbigE
(T phi, T m) Get the incomplete elliptic integral of the second kind E(φ, m).static double
bigF
(double phi, double m) Get the incomplete elliptic integral of the first kind F(φ, m).static Complex
Get the incomplete elliptic integral of the first kind F(φ, m).static Complex
bigF
(Complex phi, Complex m, ComplexUnivariateIntegrator integrator, int maxEval) Get the incomplete elliptic integral of the first kind F(φ, m) using numerical integration.static <T extends CalculusFieldElement<T>>
FieldComplex<T>bigF
(FieldComplex<T> phi, FieldComplex<T> m) Get the incomplete elliptic integral of the first kind F(φ, m).static <T extends CalculusFieldElement<T>>
FieldComplex<T>bigF
(FieldComplex<T> phi, FieldComplex<T> m, FieldComplexUnivariateIntegrator<T> integrator, int maxEval) Get the incomplete elliptic integral of the first kind F(φ, m).static <T extends CalculusFieldElement<T>>
TbigF
(T phi, T m) Get the incomplete elliptic integral of the first kind F(φ, m).static double
bigK
(double m) Get the complete elliptic integral of the first kind K(m).static Complex
Get the complete elliptic integral of the first kind K(m).static <T extends CalculusFieldElement<T>>
FieldComplex<T>bigK
(FieldComplex<T> m) Get the complete elliptic integral of the first kind K(m).static <T extends CalculusFieldElement<T>>
TbigK
(T m) Get the complete elliptic integral of the first kind K(m).static double
bigKPrime
(double m) Get the complete elliptic integral of the first kind K'(m).static Complex
Get the complete elliptic integral of the first kind K'(m).static <T extends CalculusFieldElement<T>>
FieldComplex<T>bigKPrime
(FieldComplex<T> m) Get the complete elliptic integral of the first kind K'(m).static <T extends CalculusFieldElement<T>>
TbigKPrime
(T m) Get the complete elliptic integral of the first kind K'(m).static double
bigPi
(double n, double m) Get the complete elliptic integral of the third kind Π(n, m).static double
bigPi
(double n, double phi, double m) Get the incomplete elliptic integral of the third kind Π(n, φ, m).static Complex
Get the complete elliptic integral of the third kind Π(n, m).static Complex
Get the incomplete elliptic integral of the third kind Π(n, φ, m).static Complex
bigPi
(Complex n, Complex phi, Complex m, ComplexUnivariateIntegrator integrator, int maxEval) Get the incomplete elliptic integral of the third kind Π(n, φ, m) using numerical integration.static <T extends CalculusFieldElement<T>>
FieldComplex<T>bigPi
(FieldComplex<T> n, FieldComplex<T> m) Get the complete elliptic integral of the third kind Π(n, m).static <T extends CalculusFieldElement<T>>
FieldComplex<T>bigPi
(FieldComplex<T> n, FieldComplex<T> phi, FieldComplex<T> m) Get the incomplete elliptic integral of the third kind Π(n, φ, m).static <T extends CalculusFieldElement<T>>
FieldComplex<T>bigPi
(FieldComplex<T> n, FieldComplex<T> phi, FieldComplex<T> m, FieldComplexUnivariateIntegrator<T> integrator, int maxEval) Get the incomplete elliptic integral of the third kind Π(n, φ, m).static <T extends CalculusFieldElement<T>>
TbigPi
(T n, T m) Get the complete elliptic integral of the third kind Π(n, m).static <T extends CalculusFieldElement<T>>
TbigPi
(T n, T phi, T m) Get the incomplete elliptic integral of the third kind Π(n, φ, m).static double
nome
(double m) Get the nome q.static <T extends CalculusFieldElement<T>>
Tnome
(T m) Get the nome q.
-
Method Details
-
nome
public static double nome(double m) Get the nome q.- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- nome q
-
nome
Get the nome q.- Type Parameters:
T
- the type of the field elements- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- nome q
-
bigK
public static double bigK(double m) Get the complete elliptic integral of the first kind K(m).The complete elliptic integral of the first kind K(m) is \[ \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-m \sin^2\theta}} \] it corresponds to the real quarter-period of Jacobi elliptic functions
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the first kind K(m)
- See Also:
-
bigK
Get the complete elliptic integral of the first kind K(m).The complete elliptic integral of the first kind K(m) is \[ \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-m \sin^2\theta}} \] it corresponds to the real quarter-period of Jacobi elliptic functions
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the first kind K(m)
- See Also:
-
bigK
Get the complete elliptic integral of the first kind K(m).The complete elliptic integral of the first kind K(m) is \[ \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-m \sin^2\theta}} \] it corresponds to the real quarter-period of Jacobi elliptic functions
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the first kind K(m)
- See Also:
-
bigK
Get the complete elliptic integral of the first kind K(m).The complete elliptic integral of the first kind K(m) is \[ \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-m \sin^2\theta}} \] it corresponds to the real quarter-period of Jacobi elliptic functions
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the first kind K(m)
- See Also:
-
bigKPrime
public static double bigKPrime(double m) Get the complete elliptic integral of the first kind K'(m).The complete elliptic integral of the first kind K'(m) is \[ \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-(1-m) \sin^2\theta}} \] it corresponds to the imaginary quarter-period of Jacobi elliptic functions
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the first kind K'(m)
- See Also:
-
bigKPrime
Get the complete elliptic integral of the first kind K'(m).The complete elliptic integral of the first kind K'(m) is \[ \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-(1-m) \sin^2\theta}} \] it corresponds to the imaginary quarter-period of Jacobi elliptic functions
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the first kind K'(m)
- See Also:
-
bigKPrime
Get the complete elliptic integral of the first kind K'(m).The complete elliptic integral of the first kind K'(m) is \[ \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-(1-m) \sin^2\theta}} \] it corresponds to the imaginary quarter-period of Jacobi elliptic functions
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the first kind K'(m)
- See Also:
-
bigKPrime
Get the complete elliptic integral of the first kind K'(m).The complete elliptic integral of the first kind K'(m) is \[ \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-(1-m) \sin^2\theta}} \] it corresponds to the imaginary quarter-period of Jacobi elliptic functions
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the first kind K'(m)
- See Also:
-
bigE
public static double bigE(double m) Get the complete elliptic integral of the second kind E(m).The complete elliptic integral of the second kind E(m) is \[ \int_0^{\frac{\pi}{2}} \sqrt{1-m \sin^2\theta} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the second kind E(m)
- See Also:
-
bigE
Get the complete elliptic integral of the second kind E(m).The complete elliptic integral of the second kind E(m) is \[ \int_0^{\frac{\pi}{2}} \sqrt{1-m \sin^2\theta} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the second kind E(m)
- See Also:
-
bigE
Get the complete elliptic integral of the second kind E(m).The complete elliptic integral of the second kind E(m) is \[ \int_0^{\frac{\pi}{2}} \sqrt{1-m \sin^2\theta} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the second kind E(m)
- See Also:
-
bigE
Get the complete elliptic integral of the second kind E(m).The complete elliptic integral of the second kind E(m) is \[ \int_0^{\frac{\pi}{2}} \sqrt{1-m \sin^2\theta} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the second kind E(m)
- See Also:
-
bigD
public static double bigD(double m) Get the complete elliptic integral D(m) = [K(m) - E(m)]/m.The complete elliptic integral D(m) is \[ \int_0^{\frac{\pi}{2}} \frac{\sin^2\theta}{\sqrt{1-m \sin^2\theta}} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral D(m)
- See Also:
-
bigD
Get the complete elliptic integral D(m) = [K(m) - E(m)]/m.The complete elliptic integral D(m) is \[ \int_0^{\frac{\pi}{2}} \frac{\sin^2\theta}{\sqrt{1-m \sin^2\theta}} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral D(m)
- See Also:
-
bigD
Get the complete elliptic integral D(m) = [K(m) - E(m)]/m.The complete elliptic integral D(m) is \[ \int_0^{\frac{\pi}{2}} \frac{\sin^2\theta}{\sqrt{1-m \sin^2\theta}} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral D(m)
- See Also:
-
bigD
Get the complete elliptic integral D(m) = [K(m) - E(m)]/m.The complete elliptic integral D(m) is \[ \int_0^{\frac{\pi}{2}} \frac{\sin^2\theta}{\sqrt{1-m \sin^2\theta}} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral D(m)
- See Also:
-
bigPi
public static double bigPi(double n, double m) Get the complete elliptic integral of the third kind Π(n, m).The complete elliptic integral of the third kind Π(n, m) is \[ \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-m \sin^2\theta}(1-n \sin^2\theta)} \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
n
- elliptic characteristicm
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the third kind Π(n, m)
- See Also:
-
bigPi
Get the complete elliptic integral of the third kind Π(n, m).The complete elliptic integral of the third kind Π(n, m) is \[ \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-m \sin^2\theta}(1-n \sin^2\theta)} \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
n
- elliptic characteristicm
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the third kind Π(n, m)
- See Also:
-
bigPi
Get the complete elliptic integral of the third kind Π(n, m).The complete elliptic integral of the third kind Π(n, m) is \[ \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-m \sin^2\theta}(1-n \sin^2\theta)} \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
n
- elliptic characteristicm
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the third kind Π(n, m)
- See Also:
-
bigPi
public static <T extends CalculusFieldElement<T>> FieldComplex<T> bigPi(FieldComplex<T> n, FieldComplex<T> m) Get the complete elliptic integral of the third kind Π(n, m).The complete elliptic integral of the third kind Π(n, m) is \[ \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-m \sin^2\theta}(1-n \sin^2\theta)} \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
n
- elliptic characteristicm
- parameter (m=k² where k is the elliptic modulus)- Returns:
- complete elliptic integral of the third kind Π(n, m)
- See Also:
-
bigF
public static double bigF(double phi, double m) Get the incomplete elliptic integral of the first kind F(φ, m).The incomplete elliptic integral of the first kind F(φ, m) is \[ \int_0^{\phi} \frac{d\theta}{\sqrt{1-m \sin^2\theta}} \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral of the first kind F(φ, m)
- See Also:
-
bigF
Get the incomplete elliptic integral of the first kind F(φ, m).The incomplete elliptic integral of the first kind F(φ, m) is \[ \int_0^{\phi} \frac{d\theta}{\sqrt{1-m \sin^2\theta}} \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral of the first kind F(φ, m)
- See Also:
-
bigF
Get the incomplete elliptic integral of the first kind F(φ, m).BEWARE! Elliptic integrals for complex numbers in the incomplete case are considered experimental for now, they have known issues.
The incomplete elliptic integral of the first kind F(φ, m) is \[ \int_0^{\phi} \frac{d\theta}{\sqrt{1-m \sin^2\theta}} \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral of the first kind F(φ, m)
- See Also:
-
bigF
public static Complex bigF(Complex phi, Complex m, ComplexUnivariateIntegrator integrator, int maxEval) Get the incomplete elliptic integral of the first kind F(φ, m) using numerical integration.BEWARE! Elliptic integrals for complex numbers in the incomplete case are considered experimental for now, they have known issues.
The incomplete elliptic integral of the first kind F(φ, m) is \[ \int_0^{\phi} \frac{d\theta}{\sqrt{1-m \sin^2\theta}} \]
The algorithm for evaluating the functions is based on numerical integration. If integration path comes too close to a pole of the integrand, then integration will fail with a
MathIllegalStateException
even for very largemaxEval
. This is normal behavior.- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)integrator
- integrator to usemaxEval
- maximum number of evaluations (real and imaginary parts are evaluated separately, so up to twice this number may be used)- Returns:
- incomplete elliptic integral of the first kind F(φ, m)
- See Also:
-
bigF
public static <T extends CalculusFieldElement<T>> FieldComplex<T> bigF(FieldComplex<T> phi, FieldComplex<T> m) Get the incomplete elliptic integral of the first kind F(φ, m).BEWARE! Elliptic integrals for complex numbers in the incomplete case are considered experimental for now, they have known issues.
The incomplete elliptic integral of the first kind F(φ, m) is \[ \int_0^{\phi} \frac{d\theta}{\sqrt{1-m \sin^2\theta}} \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral of the first kind F(φ, m)
- See Also:
-
bigF
public static <T extends CalculusFieldElement<T>> FieldComplex<T> bigF(FieldComplex<T> phi, FieldComplex<T> m, FieldComplexUnivariateIntegrator<T> integrator, int maxEval) Get the incomplete elliptic integral of the first kind F(φ, m).BEWARE! Elliptic integrals for complex numbers in the incomplete case are considered experimental for now, they have known issues.
The incomplete elliptic integral of the first kind F(φ, m) is \[ \int_0^{\phi} \frac{d\theta}{\sqrt{1-m \sin^2\theta}} \]
The algorithm for evaluating the functions is based on numerical integration. If integration path comes too close to a pole of the integrand, then integration will fail with a
MathIllegalStateException
even for very largemaxEval
. This is normal behavior.- Type Parameters:
T
- the type of the field elements- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)integrator
- integrator to usemaxEval
- maximum number of evaluations (real and imaginary parts are evaluated separately, so up to twice this number may be used)- Returns:
- incomplete elliptic integral of the first kind F(φ, m)
- See Also:
-
bigE
public static double bigE(double phi, double m) Get the incomplete elliptic integral of the second kind E(φ, m).The incomplete elliptic integral of the second kind E(φ, m) is \[ \int_0^{\phi} \sqrt{1-m \sin^2\theta} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral of the second kind E(φ, m)
- See Also:
-
bigE
Get the incomplete elliptic integral of the second kind E(φ, m).The incomplete elliptic integral of the second kind E(φ, m) is \[ \int_0^{\phi} \sqrt{1-m \sin^2\theta} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral of the second kind E(φ, m)
- See Also:
-
bigE
Get the incomplete elliptic integral of the second kind E(φ, m).BEWARE! Elliptic integrals for complex numbers in the incomplete case are considered experimental for now, they have known issues.
The incomplete elliptic integral of the second kind E(φ, m) is \[ \int_0^{\phi} \sqrt{1-m \sin^2\theta} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral of the second kind E(φ, m)
- See Also:
-
bigE
public static Complex bigE(Complex phi, Complex m, ComplexUnivariateIntegrator integrator, int maxEval) Get the incomplete elliptic integral of the second kind E(φ, m) using numerical integration.BEWARE! Elliptic integrals for complex numbers in the incomplete case are considered experimental for now, they have known issues.
The incomplete elliptic integral of the second kind E(φ, m) is \[ \int_0^{\phi} \sqrt{1-m \sin^2\theta} d\theta \]
The algorithm for evaluating the functions is based on numerical integration. If integration path comes too close to a pole of the integrand, then integration will fail with a
MathIllegalStateException
even for very largemaxEval
. This is normal behavior.- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)integrator
- integrator to usemaxEval
- maximum number of evaluations (real and imaginary parts are evaluated separately, so up to twice this number may be used)- Returns:
- incomplete elliptic integral of the second kind E(φ, m)
- See Also:
-
bigE
public static <T extends CalculusFieldElement<T>> FieldComplex<T> bigE(FieldComplex<T> phi, FieldComplex<T> m) Get the incomplete elliptic integral of the second kind E(φ, m).BEWARE! Elliptic integrals for complex numbers in the incomplete case are considered experimental for now, they have known issues.
The incomplete elliptic integral of the second kind E(φ, m) is \[ \int_0^{\phi} \sqrt{1-m \sin^2\theta} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral of the second kind E(φ, m)
- See Also:
-
bigE
public static <T extends CalculusFieldElement<T>> FieldComplex<T> bigE(FieldComplex<T> phi, FieldComplex<T> m, FieldComplexUnivariateIntegrator<T> integrator, int maxEval) Get the incomplete elliptic integral of the second kind E(φ, m).BEWARE! Elliptic integrals for complex numbers in the incomplete case are considered experimental for now, they have known issues.
The incomplete elliptic integral of the second kind E(φ, m) is \[ \int_0^{\phi} \sqrt{1-m \sin^2\theta} d\theta \]
The algorithm for evaluating the functions is based on numerical integration. If integration path comes too close to a pole of the integrand, then integration will fail with a
MathIllegalStateException
even for very largemaxEval
. This is normal behavior.- Type Parameters:
T
- the type of the field elements- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)integrator
- integrator to usemaxEval
- maximum number of evaluations (real and imaginary parts are evaluated separately, so up to twice this number may be used)- Returns:
- incomplete elliptic integral of the second kind E(φ, m)
- See Also:
-
bigD
public static double bigD(double phi, double m) Get the incomplete elliptic integral D(φ, m) = [F(φ, m) - E(φ, m)]/m.The incomplete elliptic integral D(φ, m) is \[ \int_0^{\phi} \frac{\sin^2\theta}{\sqrt{1-m \sin^2\theta}} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral D(φ, m)
- See Also:
-
bigD
Get the incomplete elliptic integral D(φ, m) = [F(φ, m) - E(φ, m)]/m.The incomplete elliptic integral D(φ, m) is \[ \int_0^{\phi} \frac{\sin^2\theta}{\sqrt{1-m \sin^2\theta}} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral D(φ, m)
- See Also:
-
bigD
Get the incomplete elliptic integral D(φ, m) = [F(φ, m) - E(φ, m)]/m.BEWARE! Elliptic integrals for complex numbers in the incomplete case are considered experimental for now, they have known issues.
The incomplete elliptic integral D(φ, m) is \[ \int_0^{\phi} \frac{\sin^2\theta}{\sqrt{1-m \sin^2\theta}} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral D(φ, m)
- See Also:
-
bigD
public static <T extends CalculusFieldElement<T>> FieldComplex<T> bigD(FieldComplex<T> phi, FieldComplex<T> m) Get the incomplete elliptic integral D(φ, m) = [F(φ, m) - E(φ, m)]/m.BEWARE! Elliptic integrals for complex numbers in the incomplete case are considered experimental for now, they have known issues.
The incomplete elliptic integral D(φ, m) is \[ \int_0^{\phi} \frac{\sin^2\theta}{\sqrt{1-m \sin^2\theta}} d\theta \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
phi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral D(φ, m)
- See Also:
-
bigPi
public static double bigPi(double n, double phi, double m) Get the incomplete elliptic integral of the third kind Π(n, φ, m).The incomplete elliptic integral of the third kind Π(n, φ, m) is \[ \int_0^{\phi} \frac{d\theta}{\sqrt{1-m \sin^2\theta}(1-n \sin^2\theta)} \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
n
- elliptic characteristicphi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral of the third kind Π(n, φ, m)
- See Also:
-
bigPi
Get the incomplete elliptic integral of the third kind Π(n, φ, m).The incomplete elliptic integral of the third kind Π(n, φ, m) is \[ \int_0^{\phi} \frac{d\theta}{\sqrt{1-m \sin^2\theta}(1-n \sin^2\theta)} \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
n
- elliptic characteristicphi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral of the third kind Π(n, φ, m)
- See Also:
-
bigPi
Get the incomplete elliptic integral of the third kind Π(n, φ, m).BEWARE! Elliptic integrals for complex numbers in the incomplete case are considered experimental for now, they have known issues.
The incomplete elliptic integral of the third kind Π(n, φ, m) is \[ \int_0^{\phi} \frac{d\theta}{\sqrt{1-m \sin^2\theta}(1-n \sin^2\theta)} \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Parameters:
n
- elliptic characteristicphi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral of the third kind Π(n, φ, m)
- See Also:
-
bigPi
public static Complex bigPi(Complex n, Complex phi, Complex m, ComplexUnivariateIntegrator integrator, int maxEval) Get the incomplete elliptic integral of the third kind Π(n, φ, m) using numerical integration.BEWARE! Elliptic integrals for complex numbers in the incomplete case are considered experimental for now, they have known issues.
The incomplete elliptic integral of the third kind Π(n, φ, m) is \[ \int_0^{\phi} \frac{d\theta}{\sqrt{1-m \sin^2\theta}(1-n \sin^2\theta)} \]
The algorithm for evaluating the functions is based on numerical integration. If integration path comes too close to a pole of the integrand, then integration will fail with a
MathIllegalStateException
even for very largemaxEval
. This is normal behavior.- Parameters:
n
- elliptic characteristicphi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)integrator
- integrator to usemaxEval
- maximum number of evaluations (real and imaginary- Returns:
- incomplete elliptic integral of the third kind Π(n, φ, m)
- See Also:
-
bigPi
public static <T extends CalculusFieldElement<T>> FieldComplex<T> bigPi(FieldComplex<T> n, FieldComplex<T> phi, FieldComplex<T> m) Get the incomplete elliptic integral of the third kind Π(n, φ, m).BEWARE! Elliptic integrals for complex numbers in the incomplete case are considered experimental for now, they have known issues.
The incomplete elliptic integral of the third kind Π(n, φ, m) is \[ \int_0^{\phi} \frac{d\theta}{\sqrt{1-m \sin^2\theta}(1-n \sin^2\theta)} \]
The algorithm for evaluating the functions is based on
Carlson elliptic integrals
.- Type Parameters:
T
- the type of the field elements- Parameters:
n
- elliptic characteristicphi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)- Returns:
- incomplete elliptic integral of the third kind Π(n, φ, m)
- See Also:
-
bigPi
public static <T extends CalculusFieldElement<T>> FieldComplex<T> bigPi(FieldComplex<T> n, FieldComplex<T> phi, FieldComplex<T> m, FieldComplexUnivariateIntegrator<T> integrator, int maxEval) Get the incomplete elliptic integral of the third kind Π(n, φ, m).BEWARE! Elliptic integrals for complex numbers in the incomplete case are considered experimental for now, they have known issues.
The incomplete elliptic integral of the third kind Π(n, φ, m) is \[ \int_0^{\phi} \frac{d\theta}{\sqrt{1-m \sin^2\theta}(1-n \sin^2\theta)} \]
The algorithm for evaluating the functions is based on numerical integration. If integration path comes too close to a pole of the integrand, then integration will fail with a
MathIllegalStateException
even for very largemaxEval
. This is normal behavior.- Type Parameters:
T
- the type of the field elements- Parameters:
n
- elliptic characteristicphi
- amplitude (i.e. upper bound of the integral)m
- parameter (m=k² where k is the elliptic modulus)integrator
- integrator to usemaxEval
- maximum number of evaluations (real and imaginary parts are evaluated separately, so up to twice this number may be used)- Returns:
- incomplete elliptic integral of the third kind Π(n, φ, m)
- See Also:
-