# 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 Details

• ### rC

public static double rC(double x, double y)
Compute Carlson elliptic integral RC.

The Carlson elliptic integral RCis 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 RC
• ### rC

public static <T extends CalculusFieldElement<T>> T rC(T x, T y)
Compute Carlson elliptic integral RC.

The Carlson elliptic integral RCis 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 RC
• ### rC

public static Complex rC(Complex x, Complex y)
Compute Carlson elliptic integral RC.

The Carlson elliptic integral RCis 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 RC
• ### rC

public static <T extends CalculusFieldElement<T>> FieldComplex<T> rC(FieldComplex<T> x, FieldComplex<T> y)
Compute Carlson elliptic integral RC.

The Carlson elliptic integral RCis 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 RC
• ### rF

public static double rF(double x, double y, double z)
Compute Carlson elliptic integral RF.

The Carlson elliptic integral RF 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 RF
• ### rF

public static <T extends CalculusFieldElement<T>> T rF(T x, T y, T z)
Compute Carlson elliptic integral RF.

The Carlson elliptic integral RF 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 RF
• ### rF

public static Complex rF(Complex x, Complex y, Complex z)
Compute Carlson elliptic integral RF.

The Carlson elliptic integral RF 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 RF
• ### rF

public static <T extends CalculusFieldElement<T>> FieldComplex<T> rF(FieldComplex<T> x, FieldComplex<T> y, FieldComplex<T> z)
Compute Carlson elliptic integral RF.

The Carlson elliptic integral RF 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 RF
• ### rJ

public static double rJ(double x, double y, double z, double p)
Compute Carlson elliptic integral RJ.

The Carlson elliptic integral RJ 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 RJ
• ### rJ

public static double rJ(double x, double y, double z, double p, double delta)
Compute Carlson elliptic integral RJ.

The Carlson elliptic integral RJ 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 RJ
• ### rJ

public static <T extends CalculusFieldElement<T>> T rJ(T x, T y, T z, T p)
Compute Carlson elliptic integral RJ.

The Carlson elliptic integral RJ 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 RJ
• ### rJ

public static <T extends CalculusFieldElement<T>> T rJ(T x, T y, T z, T p, T delta)
Compute Carlson elliptic integral RJ.

The Carlson elliptic integral RJ 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 RJ
• ### rJ

public static Complex rJ(Complex x, Complex y, Complex z, Complex p)
Compute Carlson elliptic integral RJ.

The Carlson elliptic integral RJ 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 RJ
• ### rJ

public static Complex rJ(Complex x, Complex y, Complex z, Complex p, Complex delta)
Compute Carlson elliptic integral RJ.

The Carlson elliptic integral RJ 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 RJ
• ### 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 RJ.

The Carlson elliptic integral RJ 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 RJ
• ### 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 RJ.

The Carlson elliptic integral RJ 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 RJ
• ### rD

public static double rD(double x, double y, double z)
Compute Carlson elliptic integral RD.

The Carlson elliptic integral RD 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 RD
• ### rD

public static <T extends CalculusFieldElement<T>> T rD(T x, T y, T z)
Compute Carlson elliptic integral RD.

The Carlson elliptic integral RD 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 RD
• ### rD

public static Complex rD(Complex x, Complex y, Complex z)
Compute Carlson elliptic integral RD.

The Carlson elliptic integral RD 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 RD
• ### rD

public static <T extends CalculusFieldElement<T>> FieldComplex<T> rD(FieldComplex<T> x, FieldComplex<T> y, FieldComplex<T> z)
Compute Carlson elliptic integral RD.

The Carlson elliptic integral RD 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 RD
• ### rG

public static double rG(double x, double y, double z)
Compute Carlson elliptic integral RG.

The Carlson elliptic integral RGis 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 RG
• ### rG

public static <T extends CalculusFieldElement<T>> T rG(T x, T y, T z)
Compute Carlson elliptic integral RG.

The Carlson elliptic integral RGis 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 RG
• ### rG

public static Complex rG(Complex x, Complex y, Complex z)
Compute Carlson elliptic integral RG.

The Carlson elliptic integral RGis 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 RG
• ### rG

public static <T extends CalculusFieldElement<T>> FieldComplex<T> rG(FieldComplex<T> x, FieldComplex<T> y, FieldComplex<T> z)
Compute Carlson elliptic integral RG.

The Carlson elliptic integral RGis 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 RG