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CarlsonRG[x,y,z]

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

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Differentiation and Integration  
Function Representations  
Function Identities and Simplifications  
Applications  
Properties & Relations  
See Also
Related Guides
History
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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:

Wolfram Language code: CarlsonRG[3., 5., 11.]

Plot over a range of arguments:

Wolfram Language code: Plot[{CarlsonRG[t, 2, 3], CarlsonRG[1, t, 3], CarlsonRG[1, 2, t]}, {t, 0, 6}, PlotLegends -> "Expressions"]

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

Wolfram Language code: 2 Sin[ϕ]CarlsonRG[Cos[ϕ]^2, 1 - m Sin[ϕ]^2, 1] /. {m -> 0.3, ϕ -> Pi / 5.}
Wolfram Language code: Sin[ϕ]^2EllipticE[ϕ, m] + Cos[ϕ]^2EllipticF[ϕ, m] + Sqrt[1 - m Sin[ϕ]^2]Sin[ϕ]Cos[ϕ] /. {m -> 0.3, ϕ -> Pi / 5}

Scope  (17)

Numerical Evaluation  (6)

Evaluate numerically:

Wolfram Language code: CarlsonRG[5, 1.0, 7]

Evaluate at high precision:

Wolfram Language code: N[CarlsonRG[3 / 4, 4, 5], 50]

Precision of the output tracks the precision of the input:

Wolfram Language code: CarlsonRG[4, 7, 1.234567890123456789012345]
Wolfram Language code: CarlsonRG[4, 7, 1.2345678901234567890123456789012345]

Evaluate for complex arguments:

Wolfram Language code: CarlsonRG[I, 1 - 2I, 3. + I]

Evaluate efficiently at high precision:

Wolfram Language code: Timing[CarlsonRG[2, 3, 7`500]]
Wolfram Language code: Timing[Precision[CarlsonRG[2, 3, 7`100000]]]

CarlsonRG threads elementwise over lists:

Wolfram Language code: CarlsonRG[{1., 2., 3.}, {2., 3., 1.}, {3., 1., 2.}]

CarlsonRG can be used with Interval and CenteredInterval objects:

Wolfram Language code: CarlsonRG[Interval[{1.23, 1.24}], Interval[{2.34, 2.35}], Interval[{3.45, 3.46}]]
Wolfram Language code: CarlsonRG[CenteredInterval[3 / 2, 1 / 100], CenteredInterval[5 / 4, 1 / 100], CenteredInterval[7 / 6, 1 / 100]]

Specific Values  (4)

Simple exact results are generated automatically:

Wolfram Language code: CarlsonRG[x, x, x]
Wolfram Language code: CarlsonRG[x, x, y]

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

Wolfram Language code: CarlsonRG[0, x, y]

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:

Wolfram Language code: CarlsonRG[x, 2 + 3I, 2 + 3I]

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

Wolfram Language code: CarlsonRG[2 + 3I, 2 + 3I, 2 + 3I]

Differentiation and Integration  (2)

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

Wolfram Language code: D[CarlsonRG[x, y, z], z]

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

Wolfram Language code: Integrate[CarlsonRG[x, y, z], z]

Function Representations  (1)

TraditionalForm formatting:

Wolfram Language code: CarlsonRG[x, y, z]//TraditionalForm

Function Identities and Simplifications  (4)

An equation relating CarlsonRG, CarlsonRF and CarlsonRD:

Wolfram Language code: 2CarlsonRG[x, y, z] == z CarlsonRF[x, y, z] + (Sqrt[x]Sqrt[y]/Sqrt[z]) - (1/3)(z - x)(z - y)CarlsonRD[x, y, z]//FullSimplify

CarlsonRG satisfies the Euler–Poisson partial differential equation:

Wolfram Language code: (x - y)Subscript[∂, x⁣y]CarlsonRG[x, y, z] + (1/2) Subscript[∂, y]CarlsonRG[x, y, z] - (1/2)Subscript[∂, x]CarlsonRG[x, y, z] == 0//FunctionExpand//FullSimplify

CarlsonRG satisfies Euler's homogeneity relation:

Wolfram Language code: x Subscript[∂, x]CarlsonRG[x, y, z] + y Subscript[∂, y]CarlsonRG[x, y, z] + z Subscript[∂, z]CarlsonRG[x, y, z] == (1/2)CarlsonRG[x, y, z]//FunctionExpand//FullSimplify

A partial differential equation satisfied by CarlsonRG:

Wolfram Language code: Subscript[∂, x]CarlsonRG[x, y, z] + Subscript[∂, y]CarlsonRG[x, y, z] + Subscript[∂, z]CarlsonRG[x, y, z] == (1/2)CarlsonRF[x, y, z]//FunctionExpand//FullSimplify

Applications  (2)

Calculate the surface area of a triaxial ellipsoid:

Wolfram Language code: area[a_, b_, c_] := 4π a b c CarlsonRG[(1/a^2), (1/b^2), (1/c^2)];

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

Wolfram Language code: area[3, 2, 1]//N

Use RegionMeasure to calculate the surface area of the ellipsoid:

Wolfram Language code: RegionMeasure[RegionBoundary[Ellipsoid[{0, 0, 0}, {3, 2, 1}]], WorkingPrecision -> MachinePrecision]

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

Wolfram Language code: MatrixForm[mat = HilbertMatrix[3]]
Wolfram Language code: NExpectation[Sqrt[(1/2){u, v, w}.mat.{u, v, w}], {u, v, w}ProductDistribution[{NormalDistribution[], 3}], Method -> "MonteCarlo"]//Quiet

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

Wolfram Language code: N[(Gamma[(3 - 1/2)]/Gamma[(3/2)])(CarlsonRG@@Eigenvalues[mat]), 25]

Properties & Relations  (1)

CarlsonRG is invariant under a permutation of its arguments:

Wolfram Language code: CarlsonRG[x, y, z] == CarlsonRG[z, x, y]
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_2026_carlsonrg, author="Wolfram Research", title="{CarlsonRG}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRG.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_carlsonrg, organization={Wolfram Research}, title={CarlsonRG}, year={2023}, url={https://reference.wolfram.com/language/ref/CarlsonRG.html}, note=[Accessed: 12-June-2026]}