# CarlsonRJ

CarlsonRJ[x,y,z,ρ]

gives Carlson's elliptic integral .

# Details

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

# Examples

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## Basic Examples(3)

Evaluate numerically:

Visualize over a range of arguments:

CarlsonRJ is related to the Legendre elliptic integral of the third kind for :

## Scope(13)

### 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:

CarlsonRJ can be used with Interval and CenteredInterval objects:

### Specific Values(3)

Simple exact values are generated automatically:

When one of the first three arguments of CarlsonRJ is zero, CarlsonRJ reduces to the complete elliptic integral CarlsonRM:

When one of the first three arguments of CarlsonRJ is equal to the last argument, and they do not lie on the negative real axis, CarlsonRJ reduces to CarlsonRD:

### Derivatives(1)

Derivative of with respect to :

Derivative of with respect to :

### Function Identities and Simplifications(2)

CarlsonRJ satisfies the EulerPoisson partial differential equation:

CarlsonRJ satisfies Euler's homogeneity relation:

## Applications(1)

Use CarlsonRJ to define a conformal map:

Visualize the image of lines of constant real and imaginary parts:

## Properties & Relations(1)

CarlsonRJ is invariant under a permutation of its first three arguments:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_carlsonrj, author="Wolfram Research", title="{CarlsonRJ}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRJ.html}", note=[Accessed: 22-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_carlsonrj, organization={Wolfram Research}, title={CarlsonRJ}, year={2023}, url={https://reference.wolfram.com/language/ref/CarlsonRJ.html}, note=[Accessed: 22-May-2024 ]}