gives the Carlson's elliptic integral .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For non-negative arguments, .
  • 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.


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Basic Examples  (3)

Evaluate numerically:

Plot over a range of arguments:

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

Scope  (9)

Numerical Evaluation  (5)

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:

Specific Values  (2)

Simple exact results are generated automatically:

The case of complete elliptic integral of the second kind:

Derivatives  (1)

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

Function Representations  (1)

TraditionalForm formatting:

Applications  (2)

Calculate the surface area of a triaxial ellipsoid:

The area of an ellipsoid with semiaxes 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 to closed-form result in terms of CarlsonRG:

Wolfram Research (2021), CarlsonRG, Wolfram Language function,


Wolfram Research (2021), CarlsonRG, Wolfram Language function,


Wolfram Language. 2021. "CarlsonRG." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2021). CarlsonRG. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2022_carlsonrg, author="Wolfram Research", title="{CarlsonRG}", year="2021", howpublished="\url{}", note=[Accessed: 03-June-2023 ]}


@online{reference.wolfram_2022_carlsonrg, organization={Wolfram Research}, title={CarlsonRG}, year={2021}, url={}, note=[Accessed: 03-June-2023 ]}