CarlsonRF

CarlsonRF[x,y,z]

gives the Carlson's elliptic integral TemplateBox[{x, y, z}, CarlsonRF].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For , and , TemplateBox[{x, y, z}, CarlsonRF]⩵1/2int_0^infty(t+x)^(-1/2)(t+y)^(-1/2)(t+z)^(-1/2)dt.
  • CarlsonRF[x,y,z] has a branch cut discontinuity for .
  • For certain arguments, CarlsonRF automatically evaluates to exact values.
  • CarlsonRF can be evaluated to arbitrary numerical precision.
  • CarlsonRF automatically threads over lists.
  • CarlsonRF can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (3)

Evaluate numerically:

Plot over a range of arguments:

CarlsonRF is related to the Legendre elliptic integral of the first kind TemplateBox[{phi, m}, EllipticF] for :

Scope  (17)

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:

CarlsonRF threads elementwise over lists:

CarlsonRF can be used with Interval and CenteredInterval objects:

Specific Values  (4)

Simple exact results are generated automatically:

When one argument of CarlsonRF is zero, CarlsonRF reduces to the complete elliptic integral CarlsonRK:

When two of the arguments of CarlsonRF are identical and do not lie on the negative real axis, CarlsonRF reduces to CarlsonRC:

When all arguments of CarlsonRF are identical and do not lie on the negative real axis, CarlsonRF reduces to an elementary function:

Differentiation and Integration  (2)

Derivative of CarlsonRF is proportional to CarlsonRD:

Indefinite integral of CarlsonRF:

Function Representations  (1)

TraditionalForm formatting:

Function Identities and Simplifications  (4)

An equation relating CarlsonRF, CarlsonRG and CarlsonRD:

CarlsonRF satisfies the Euler–Poisson partial differential equation:

CarlsonRF satisfies Euler's homogeneity relation:

A partial differential equation satisfied by CarlsonRF:

Applications  (3)

Distance along a meridian of the Earth:

Compare with the result of GeoDistance:

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

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

Express EllipticLog in terms of CarlsonRF:

Properties & Relations  (1)

CarlsonRF is invariant under a permutation of its arguments:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_carlsonrf, author="Wolfram Research", title="{CarlsonRF}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRF.html}", note=[Accessed: 06-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_carlsonrf, organization={Wolfram Research}, title={CarlsonRF}, year={2023}, url={https://reference.wolfram.com/language/ref/CarlsonRF.html}, note=[Accessed: 06-May-2025 ]}