# CarlsonRF

CarlsonRF[x,y,z]

gives the Carlson's elliptic integral .

# Details

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

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot over a range of arguments:

CarlsonRF is related to the Legendre elliptic integral of the first 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:

CarlsonRF can be used with Interval and CenteredInterval objects:

### Specific Values(4)

Simple exact results are generated automatically:

When one argument of CarlsonRF is zero, CarlsonRF reduces to the complete elliptic integral CarlsonRK:

When two of the arguments of CarlsonRF are identical and do not lie on the negative real axis, CarlsonRF reduces to CarlsonRC:

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

### Derivatives(1)

Derivative of CarlsonRF is proportional to CarlsonRD:

### Function Identities and Simplifications(4)

An equation relating CarlsonRF, CarlsonRG and CarlsonRD:

CarlsonRF satisfies the EulerPoisson partial differential equation:

CarlsonRF satisfies Euler's homogeneity relation:

A partial differential equation satisfied by CarlsonRF:

## Applications(3)

Distance along a meridian of the Earth:

Compare with the result of GeoDistance:

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

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

## Properties & Relations(1)

CarlsonRF is invariant under a permutation of its arguments:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_carlsonrf, author="Wolfram Research", title="{CarlsonRF}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRF.html}", note=[Accessed: 29-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_carlsonrf, organization={Wolfram Research}, title={CarlsonRF}, year={2023}, url={https://reference.wolfram.com/language/ref/CarlsonRF.html}, note=[Accessed: 29-May-2024 ]}