gives the Carlson's elliptic integral .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For , and , .
  • 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.


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Basic Examples  (3)

Evaluate numerically:

Plot over a range of arguments:

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

CarlsonRF threads elementwise over lists:

Specific Values  (2)

Simple exact results are generated automatically:

The case of complete elliptic integral:

Derivatives  (1)

Derivative of CarlsonRF is proportional to CarlsonRD:

Function Representations  (1)

TraditionalForm formatting:

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 to closed-form result in terms of CarlsonRF:

Express EllipticLog in terms of CarlsonRF:

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


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


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


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


@misc{reference.wolfram_2022_carlsonrf, author="Wolfram Research", title="{CarlsonRF}", year="2021", howpublished="\url{}", note=[Accessed: 01-June-2023 ]}


@online{reference.wolfram_2022_carlsonrf, organization={Wolfram Research}, title={CarlsonRF}, year={2021}, url={}, note=[Accessed: 01-June-2023 ]}