# CarlsonRE

CarlsonRE[x,y]

gives the Carlson's elliptic integral .

# Details

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

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot the function:

CarlsonRE is related to Legendre's complete elliptic integral of the second kind:

## Scope(11)

### Numerical Evaluation(5)

Evaluate numerically:

Evaluate to high precision:

Precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate efficiently at high precision:

CarlsonRE threads elementwise over lists:

### Specific Values(1)

Simple exact values are generated automatically:

Derivatives:

### Function Identities and Simplifications(3)

CarlsonRE satisfies the EulerPoisson partial differential equation:

CarlsonRE satisfies Euler's homogeneity relation:

A partial differential equation satisfied by CarlsonRE:

## Applications(3)

Total arc length of an ellipse:

Compare with the result of ArcLength:

Expectation value of the square root of a quadratic form, relative to a normal distribution:

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

Visualize the intersection of a cylinder and a ball:

Volume of cylinder-ball intersection expressed in terms of Carlson integrals:

Compare with the result of Volume:

## Properties & Relations(2)

CarlsonRE is invariant under a permutation of its arguments:

CarlsonRE and CarlsonRK satisfy Legendre's relation:

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

#### Text

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

#### CMS

Wolfram Language. 2021. "CarlsonRE." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CarlsonRE.html.

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_carlsonre, author="Wolfram Research", title="{CarlsonRE}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRE.html}", note=[Accessed: 22-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_carlsonre, organization={Wolfram Research}, title={CarlsonRE}, year={2021}, url={https://reference.wolfram.com/language/ref/CarlsonRE.html}, note=[Accessed: 22-May-2024 ]}