# CarlsonRD

CarlsonRD[x,y,z]

gives the Carlson's elliptic integral .

# Details

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

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot CarlsonRD:

CarlsonRD is related to a combination of Legendre elliptic integrals restricted to :

## Scope(15)

### Numerical Evaluation(6)

Evaluate CarlsonRD numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate efficiently at high precision:

CarlsonRD threads elementwise over lists:

CarlsonRD can be used with Interval and CenteredInterval objects:

### Specific Values(2)

Simple exact values are generated automatically:

When one argument of CarlsonRD is zero, CarlsonRD can be expressed in terms of the complete elliptic integrals CarlsonRE and CarlsonRK:

### Differentiation and Integration(2)

Derivative of with respect to :

Derivative of with respect to :

Indefinite integral of with respect to :

Indefinite integral of with respect to :

### Function Representations(1)

TraditionalForm formatting:

### Function Identities and Simplifications(4)

An equation relating CarlsonRD, CarlsonRF and CarlsonRG:

Some cyclic permutation identities for CarlsonRD:

CarlsonRD satisfies the EulerPoisson partial differential equation:

CarlsonRD satisfies Euler's homogeneity relation:

## Applications(2)

Distance along a meridian of the Earth:

Compare with the result of GeoDistance:

Parametrization of a Mylar balloon (two flat sheets of plastic sewn together at their circumference and then inflated):

## Properties & Relations(1)

CarlsonRD is symmetric with respect to its first two arguments:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_carlsonrd, author="Wolfram Research", title="{CarlsonRD}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRD.html}", note=[Accessed: 18-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_carlsonrd, organization={Wolfram Research}, title={CarlsonRD}, year={2023}, url={https://reference.wolfram.com/language/ref/CarlsonRD.html}, note=[Accessed: 18-September-2024 ]}