CarlsonRG

CarlsonRG[x,y,z]

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

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For non-negative arguments, TemplateBox[{x, y, z}, CarlsonRG]=1/4int_0^infty(t+x)^(-1/2)(t+y)^(-1/2)(t+z)^(-1/2)((x t)/(t+x)+(y t)/(t+y)+(z t)/(t+z))dt.
  • CarlsonRG[x,y,z] has a branch cut discontinuity at .
  • For certain arguments, CarlsonRG automatically evaluates to exact values.
  • CarlsonRG can be evaluated to arbitrary precision.
  • CarlsonRG automatically threads over lists.
  • CarlsonRG can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (3)

Evaluate numerically:

Plot over a range of arguments:

CarlsonRG is related to the Legendre elliptic integral of the second kind TemplateBox[{phi, m}, EllipticE2] 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:

CarlsonRG threads elementwise over lists:

CarlsonRG can be used with Interval and CenteredInterval objects:

Specific Values  (4)

Simple exact results are generated automatically:

When one argument of CarlsonRG is zero, CarlsonRG reduces to the complete elliptic integral CarlsonRE:

When two of the arguments of CarlsonRG are identical and do not lie on the negative real axis, CarlsonRG can be expressed in terms of CarlsonRC:

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

Differentiation and Integration  (2)

Derivative of TemplateBox[{x, y, z}, CarlsonRG] with respect to :

Indefinite integral of TemplateBox[{x, y, z}, CarlsonRG] with respect to :

Function Representations  (1)

TraditionalForm formatting:

Function Identities and Simplifications  (4)

An equation relating CarlsonRG, CarlsonRF and CarlsonRD:

CarlsonRG satisfies the Euler–Poisson partial differential equation:

CarlsonRG satisfies Euler's homogeneity relation:

A partial differential equation satisfied by CarlsonRG:

Applications  (2)

Calculate the surface area of a triaxial ellipsoid:

The area of an ellipsoid with semi‐axes 3, 2, 1:

Use RegionMeasure to calculate the surface area of the ellipsoid:

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

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

Properties & Relations  (1)

CarlsonRG is invariant under a permutation of its arguments:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2025_carlsonrg, author="Wolfram Research", title="{CarlsonRG}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRG.html}", note=[Accessed: 25-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_carlsonrg, organization={Wolfram Research}, title={CarlsonRG}, year={2023}, url={https://reference.wolfram.com/language/ref/CarlsonRG.html}, note=[Accessed: 25-May-2025 ]}