# CarlsonRM

CarlsonRM[x,y,ρ]

gives Carlson's elliptic integral .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• For non-negative arguments, .
• CarlsonRM[x,y,ρ] has a branch cut discontinuity at .
• CarlsonRM[x,y,ρ] is understood as a Cauchy principal value integral for ρ<0.
• For certain arguments, CarlsonRM automatically evaluates to exact values.
• CarlsonRM can be evaluated to arbitrary precision.
• CarlsonRM automatically threads over lists.

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot the function:

CarlsonRM is related to the complete Legendre elliptic integral of the third kind:

## Scope(10)

### Numerical Evaluation(5)

Evaluate numerically:

Evaluate numerically to high precision:

Precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate efficiently at high precision:

### Specific Values(1)

Simple exact values are generated automatically:

### Derivatives(1)

Derivative of with respect to :

Derivative of with respect to :

### Function Identities and Simplifications(2)

CarlsonRM satisfies the EulerPoisson partial differential equation:

CarlsonRM satisfies Euler's homogeneity relation:

## Applications(2)

Visualize the solid angle subtended by a circular disk:

Evaluate the solid angle:

Compare with the result of NIntegrate:

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(1)

CarlsonRM is symmetric with respect to its first two arguments:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_carlsonrm, author="Wolfram Research", title="{CarlsonRM}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRM.html}", note=[Accessed: 19-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_carlsonrm, organization={Wolfram Research}, title={CarlsonRM}, year={2021}, url={https://reference.wolfram.com/language/ref/CarlsonRM.html}, note=[Accessed: 19-June-2024 ]}