# CarlsonRG

CarlsonRG[x,y,z]

gives the Carlson's elliptic integral .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• For non-negative arguments, .
• CarlsonRG[x,y,z] has a branch cut discontinuity at .
• For certain arguments, CarlsonRG automatically evaluates to exact values.
• CarlsonRG can be evaluated to arbitrary precision.
• CarlsonRG automatically threads over lists.
• CarlsonRG can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot over a range of arguments:

CarlsonRG is related to the Legendre elliptic integral of the second kind for :

## Scope(16)

### Numerical Evaluation(6)

Evaluate numerically:

Evaluate at high precision:

Precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate efficiently at high precision:

CarlsonRG can be used with Interval and CenteredInterval objects:

### Specific Values(4)

Simple exact results are generated automatically:

When one argument of CarlsonRG is zero, CarlsonRG reduces to the complete elliptic integral CarlsonRE:

When two of the arguments of CarlsonRG are identical and do not lie on the negative real axis, CarlsonRG can be expressed in terms of CarlsonRC:

When all arguments of CarlsonRG are identical and do not lie on the negative real axis, CarlsonRG reduces to an elementary function:

### Derivatives(1)

Derivative of with respect to :

### Function Identities and Simplifications(4)

An equation relating CarlsonRG, CarlsonRF and CarlsonRD:

CarlsonRG satisfies the EulerPoisson partial differential equation:

CarlsonRG satisfies Euler's homogeneity relation:

A partial differential equation satisfied by CarlsonRG:

## Applications(2)

Calculate the surface area of a triaxial ellipsoid:

The area of an ellipsoid with semiaxes 3, 2, 1:

Use RegionMeasure to calculate the surface area of the ellipsoid:

Expectation value of the square root of a quadratic form over a normal distribution:

Compare with the closed-form result in terms of CarlsonRG:

## Properties & Relations(1)

CarlsonRG is invariant under a permutation of its arguments:

Wolfram Research (2021), CarlsonRG, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRG.html (updated 2023).

#### Text

Wolfram Research (2021), CarlsonRG, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRG.html (updated 2023).

#### CMS

Wolfram Language. 2021. "CarlsonRG." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/CarlsonRG.html.

#### APA

Wolfram Language. (2021). CarlsonRG. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CarlsonRG.html

#### BibTeX

@misc{reference.wolfram_2024_carlsonrg, author="Wolfram Research", title="{CarlsonRG}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRG.html}", note=[Accessed: 17-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_carlsonrg, organization={Wolfram Research}, title={CarlsonRG}, year={2023}, url={https://reference.wolfram.com/language/ref/CarlsonRG.html}, note=[Accessed: 17-June-2024 ]}