Package org.hipparchus.special.elliptic.carlson

Class CarlsonEllipticIntegral

java.lang.Object

org.hipparchus.special.elliptic.carlson.CarlsonEllipticIntegral

public class CarlsonEllipticIntegral
extends Object

Elliptic integrals in Carlson symmetric form.

This utility class computes the various symmetric elliptic integrals defined as: \[ \left\{\begin{align} R_F(x,y,z) &= \frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{s(t)}\\ R_J(x,y,z,p) &= \frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{s(t)(t+p)}\\ R_G(x,y,z) &= \frac{1}{4}\int_{0}^{\infty}\frac{1}{s(t)} \left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\mathrm{d}t\\ R_D(x,y,z) &= R_J(x,y,z,z)\\ R_C(x,y) &= R_F(x,y,y) \end{align}\right. \]

where \[ s(t) = \sqrt{t+x}\sqrt{t+y}\sqrt{t+z} \]

The algorithms used are based on the duplication method as described in B. C. Carlson 1995 paper "Numerical computation of real or complex elliptic integrals", with the improvements described in the appendix of B. C. Carlson and James FitzSimons 2000 paper "Reduction theorems for elliptic integrands with the square root of two quadratic factors". They are also described in section 19.36(i) of Digital Library of Mathematical 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.

Since:

2.0

Method Summary

Modifier and Type	Method	Description
static double	rC(double x, double y)	Compute Carlson elliptic integral RC.
static Complex	rC(Complex x, Complex y)	Compute Carlson elliptic integral RC.
static <T extends CalculusFieldElement<T>> FieldComplex<T>	rC(FieldComplex<T> x, FieldComplex<T> y)	Compute Carlson elliptic integral RC.
static <T extends CalculusFieldElement<T>> T	rC(T x, T y)	Compute Carlson elliptic integral RC.
static double	rD(double x, double y, double z)	Compute Carlson elliptic integral RD.
static Complex	rD(Complex x, Complex y, Complex z)	Compute Carlson elliptic integral RD.
static <T extends CalculusFieldElement<T>> FieldComplex<T>	rD(FieldComplex<T> x, FieldComplex<T> y, FieldComplex<T> z)	Compute Carlson elliptic integral RD.
static <T extends CalculusFieldElement<T>> T	rD(T x, T y, T z)	Compute Carlson elliptic integral RD.
static double	rF(double x, double y, double z)	Compute Carlson elliptic integral RF.
static Complex	rF(Complex x, Complex y, Complex z)	Compute Carlson elliptic integral RF.
static <T extends CalculusFieldElement<T>> FieldComplex<T>	rF(FieldComplex<T> x, FieldComplex<T> y, FieldComplex<T> z)	Compute Carlson elliptic integral RF.
static <T extends CalculusFieldElement<T>> T	rF(T x, T y, T z)	Compute Carlson elliptic integral RF.
static double	rG(double x, double y, double z)	Compute Carlson elliptic integral RG.
static Complex	rG(Complex x, Complex y, Complex z)	Compute Carlson elliptic integral RG.
static <T extends CalculusFieldElement<T>> FieldComplex<T>	rG(FieldComplex<T> x, FieldComplex<T> y, FieldComplex<T> z)	Compute Carlson elliptic integral RG.
static <T extends CalculusFieldElement<T>> T	rG(T x, T y, T z)	Compute Carlson elliptic integral RG.
static double	rJ(double x, double y, double z, double p)	Compute Carlson elliptic integral RJ.
static double	rJ(double x, double y, double z, double p, double delta)	Compute Carlson elliptic integral RJ.
static Complex	rJ(Complex x, Complex y, Complex z, Complex p)	Compute Carlson elliptic integral RJ.
static Complex	rJ(Complex x, Complex y, Complex z, Complex p, Complex delta)	Compute Carlson elliptic integral RJ.
static <T extends CalculusFieldElement<T>> FieldComplex<T>	rJ(FieldComplex<T> x, FieldComplex<T> y, FieldComplex<T> z, FieldComplex<T> p)	Compute Carlson elliptic integral RJ.
static <T extends CalculusFieldElement<T>> FieldComplex<T>	rJ(FieldComplex<T> x, FieldComplex<T> y, FieldComplex<T> z, FieldComplex<T> p, FieldComplex<T> delta)	Compute Carlson elliptic integral RJ.
static <T extends CalculusFieldElement<T>> T	rJ(T x, T y, T z, T p)	Compute Carlson elliptic integral RJ.
static <T extends CalculusFieldElement<T>> T	rJ(T x, T y, T z, T p, T delta)	Compute Carlson elliptic integral RJ.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Detail

rC

public static double rC(double x,
                        double y)

Compute Carlson elliptic integral R_C.

The Carlson elliptic integral R_Cis defined as \[ R_C(x,y,z)=R_F(x,y,y)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}(t+y)} \]

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

Returns:

Carlson elliptic integral R_C

rC

public static <T extends CalculusFieldElement<T>> T rC(T x,
                                                       T y)

Compute Carlson elliptic integral R_C.

The Carlson elliptic integral R_Cis defined as \[ R_C(x,y,z)=R_F(x,y,y)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}(t+y)} \]

Type Parameters:

T - type of the field elements

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

Returns:

Carlson elliptic integral R_C

rC

public static Complex rC(Complex x,
                         Complex y)

Compute Carlson elliptic integral R_C.

The Carlson elliptic integral R_Cis defined as \[ R_C(x,y,z)=R_F(x,y,y)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}(t+y)} \]

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

Returns:

Carlson elliptic integral R_C

rC

public static <T extends CalculusFieldElement<T>> FieldComplex<T> rC(FieldComplex<T> x,
                                                                     FieldComplex<T> y)

Compute Carlson elliptic integral R_C.

The Carlson elliptic integral R_Cis defined as \[ R_C(x,y,z)=R_F(x,y,y)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}(t+y)} \]

Type Parameters:

T - type of the field elements

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

Returns:

Carlson elliptic integral R_C

rF

public static double rF(double x,
                        double y,
                        double z)

Compute Carlson elliptic integral R_F.

The Carlson elliptic integral R_F is defined as \[ R_F(x,y,z)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}} \]

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

Returns:

Carlson elliptic integral R_F

rF

public static <T extends CalculusFieldElement<T>> T rF(T x,
                                                       T y,
                                                       T z)

Compute Carlson elliptic integral R_F.

The Carlson elliptic integral R_F is defined as \[ R_F(x,y,z)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}} \]

Type Parameters:

T - type of the field elements

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

Returns:

Carlson elliptic integral R_F

rF

public static Complex rF(Complex x,
                         Complex y,
                         Complex z)

Compute Carlson elliptic integral R_F.

The Carlson elliptic integral R_F is defined as \[ R_F(x,y,z)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}} \]

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

Returns:

Carlson elliptic integral R_F

rF

public static <T extends CalculusFieldElement<T>> FieldComplex<T> rF(FieldComplex<T> x,
                                                                     FieldComplex<T> y,
                                                                     FieldComplex<T> z)

Compute Carlson elliptic integral R_F.

The Carlson elliptic integral R_F is defined as \[ R_F(x,y,z)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}} \]

Type Parameters:

T - type of the field elements

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

Returns:

Carlson elliptic integral R_F

rJ

public static double rJ(double x,
                        double y,
                        double z,
                        double p)

Compute Carlson elliptic integral R_J.

The Carlson elliptic integral R_J is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

p - fourth not symmetric variable of the integral

Returns:

Carlson elliptic integral R_J

rJ

public static double rJ(double x,
                        double y,
                        double z,
                        double p,
                        double delta)

Compute Carlson elliptic integral R_J.

The Carlson elliptic integral R_J is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

p - fourth not symmetric variable of the integral

delta - precomputed value of (p-x)(p-y)(p-z)

Returns:

Carlson elliptic integral R_J

rJ

public static <T extends CalculusFieldElement<T>> T rJ(T x,
                                                       T y,
                                                       T z,
                                                       T p)

Compute Carlson elliptic integral R_J.

The Carlson elliptic integral R_J is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]

Type Parameters:

T - type of the field elements

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

p - fourth not symmetric variable of the integral

Returns:

Carlson elliptic integral R_J

rJ

public static <T extends CalculusFieldElement<T>> T rJ(T x,
                                                       T y,
                                                       T z,
                                                       T p,
                                                       T delta)

Compute Carlson elliptic integral R_J.

The Carlson elliptic integral R_J is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]

Type Parameters:

T - type of the field elements

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

p - fourth not symmetric variable of the integral

delta - precomputed value of (p-x)(p-y)(p-z)

Returns:

Carlson elliptic integral R_J

rJ

public static Complex rJ(Complex x,
                         Complex y,
                         Complex z,
                         Complex p)

Compute Carlson elliptic integral R_J.

The Carlson elliptic integral R_J is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

p - fourth not symmetric variable of the integral

Returns:

Carlson elliptic integral R_J

rJ

public static Complex rJ(Complex x,
                         Complex y,
                         Complex z,
                         Complex p,
                         Complex delta)

Compute Carlson elliptic integral R_J.

The Carlson elliptic integral R_J is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

p - fourth not symmetric variable of the integral

delta - precomputed value of (p-x)(p-y)(p-z)

Returns:

Carlson elliptic integral R_J

rJ

public static <T extends CalculusFieldElement<T>> FieldComplex<T> rJ(FieldComplex<T> x,
                                                                     FieldComplex<T> y,
                                                                     FieldComplex<T> z,
                                                                     FieldComplex<T> p)

Compute Carlson elliptic integral R_J.

The Carlson elliptic integral R_J is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]

Type Parameters:

T - type of the field elements

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

p - fourth not symmetric variable of the integral

Returns:

Carlson elliptic integral R_J

rJ

public static <T extends CalculusFieldElement<T>> FieldComplex<T> rJ(FieldComplex<T> x,
                                                                     FieldComplex<T> y,
                                                                     FieldComplex<T> z,
                                                                     FieldComplex<T> p,
                                                                     FieldComplex<T> delta)

Compute Carlson elliptic integral R_J.

The Carlson elliptic integral R_J is defined as \[ R_J(x,y,z,p)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+p)} \]

Type Parameters:

T - type of the field elements

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

p - fourth not symmetric variable of the integral

delta - precomputed value of (p-x)(p-y)(p-z)

Returns:

Carlson elliptic integral R_J

rD

public static double rD(double x,
                        double y,
                        double z)

Compute Carlson elliptic integral R_D.

The Carlson elliptic integral R_D is defined as \[ R_D(x,y,z)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+z)} \]

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

Returns:

Carlson elliptic integral R_D

rD

public static <T extends CalculusFieldElement<T>> T rD(T x,
                                                       T y,
                                                       T z)

Compute Carlson elliptic integral R_D.

The Carlson elliptic integral R_D is defined as \[ R_D(x,y,z)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+z)} \]

Type Parameters:

T - type of the field elements

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

Returns:

Carlson elliptic integral R_D

rD

public static Complex rD(Complex x,
                         Complex y,
                         Complex z)

Compute Carlson elliptic integral R_D.

The Carlson elliptic integral R_D is defined as \[ R_D(x,y,z)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+z)} \]

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

Returns:

Carlson elliptic integral R_D

rD

public static <T extends CalculusFieldElement<T>> FieldComplex<T> rD(FieldComplex<T> x,
                                                                     FieldComplex<T> y,
                                                                     FieldComplex<T> z)

Compute Carlson elliptic integral R_D.

The Carlson elliptic integral R_D is defined as \[ R_D(x,y,z)=\frac{3}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}(t+z)} \]

Type Parameters:

T - type of the field elements

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - third symmetric variable of the integral

Returns:

Carlson elliptic integral R_D

rG

public static double rG(double x,
                        double y,
                        double z)

Compute Carlson elliptic integral R_G.

The Carlson elliptic integral R_Gis defined as \[ R_{G}(x,y,z)=\frac{1}{4}\int_{0}^{\infty}\frac{1}{s(t)} \left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\mathrm{d}t \]

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - second symmetric variable of the integral

Returns:

Carlson elliptic integral R_G

rG

public static <T extends CalculusFieldElement<T>> T rG(T x,
                                                       T y,
                                                       T z)

Compute Carlson elliptic integral R_G.

The Carlson elliptic integral R_Gis defined as \[ R_{G}(x,y,z)=\frac{1}{4}\int_{0}^{\infty}\frac{1}{s(t)} \left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\mathrm{d}t \]

Type Parameters:

T - type of the field elements

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - second symmetric variable of the integral

Returns:

Carlson elliptic integral R_G

rG

public static Complex rG(Complex x,
                         Complex y,
                         Complex z)

Compute Carlson elliptic integral R_G.

The Carlson elliptic integral R_Gis defined as \[ R_{G}(x,y,z)=\frac{1}{4}\int_{0}^{\infty}\frac{1}{s(t)} \left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\mathrm{d}t \]

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - second symmetric variable of the integral

Returns:

Carlson elliptic integral R_G

rG

public static <T extends CalculusFieldElement<T>> FieldComplex<T> rG(FieldComplex<T> x,
                                                                     FieldComplex<T> y,
                                                                     FieldComplex<T> z)

Compute Carlson elliptic integral R_G.

The Carlson elliptic integral R_Gis defined as \[ R_{G}(x,y,z)=\frac{1}{4}\int_{0}^{\infty}\frac{1}{s(t)} \left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\mathrm{d}t \]

Type Parameters:

T - type of the field elements

Parameters:

x - first symmetric variable of the integral

y - second symmetric variable of the integral

z - second symmetric variable of the integral

Returns:

Carlson elliptic integral R_G