# CarlsonRK

CarlsonRK[x,y]

gives the Carlson's elliptic integral .

# Details

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

# Examples

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## Basic Examples(3)

Evaluate numerically:

Plot the function:

CarlsonRK is related to the complete elliptic integral of the first kind EllipticK:

## Scope(8)

### 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:

### Specific Values(1)

Simple exact values are generated automatically:

### Derivatives(1)

Derivative with respect to :

Derivative with respect to :

## Applications(5)

Total arc length of a lemniscate of Bernoulli:

Compare with the result of ArcLength:

Evaluate an elliptic singular value:

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

Compare to closed-form result in terms of CarlsonRK:

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:

Compare with the result of Volume:

## Properties & Relations(2)

CarlsonRE and CarlsonRK satisfy Legendre's relation:

CarlsonRK is related to ArithmeticGeometricMean:

## Neat Examples(1)

Probability that a random walker in a 3D cubic lattice returns to the origin:

Carry out a modeling run of 1000 walks and count how many return to the origin:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2022_carlsonrk, author="Wolfram Research", title="{CarlsonRK}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRK.html}", note=[Accessed: 11-August-2022 ]}

#### BibLaTeX

@online{reference.wolfram_2022_carlsonrk, organization={Wolfram Research}, title={CarlsonRK}, year={2021}, url={https://reference.wolfram.com/language/ref/CarlsonRK.html}, note=[Accessed: 11-August-2022 ]}