gives the Carlson's elliptic integral .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For and , .
  • CarlsonRC[x,y] has a branch cut discontinuity at .
  • CarlsonRC[x,y] is real valued for and , being understood as a Cauchy principal value integral for .
  • For certain arguments, CarlsonRC automatically evaluates to exact values.
  • CarlsonRC can be evaluated to arbitrary numerical precision.
  • CarlsonRC automatically threads over lists.


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Basic Examples  (3)

Evaluate numerically:

Plot the function:

CarlsonRC is related to the special case of TemplateBox[{phi, m}, EllipticF] for :

Scope  (8)

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:

CarlsonRC threads elementwise over lists:

Specific Values  (1)

Simple exact values are generated automatically:

Derivatives  (1)

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

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

Function Representations  (1)

TraditionalForm formatting:

Applications  (2)

Use CarlsonRC to provide an upper bound and lower bounds for CarlsonRF[x,y,z]:

CarlsonRC is useful for compactly expressing the change of parameter relations for EllipticPi:

Properties & Relations  (2)

For , the TemplateBox[{x, y}, CarlsonRC] can be expressed in terms of ArcCos:

TemplateBox[{x, y}, CarlsonRC] is interpreted as a principal value for on the negative real axis:

Compare to the closed-form result in terms of positive arguments:

Possible Issues  (1)

Generically, TemplateBox[{x, z, z}, CarlsonRF]=TemplateBox[{x, z}, CarlsonRC]; however, due to differences in analytic structures of the two functions, the evaluation is restricted to numeric values of that can be checked to lay off the negative real axis:

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


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


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


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


@misc{reference.wolfram_2022_carlsonrc, author="Wolfram Research", title="{CarlsonRC}", year="2021", howpublished="\url{}", note=[Accessed: 20-March-2023 ]}


@online{reference.wolfram_2022_carlsonrc, organization={Wolfram Research}, title={CarlsonRC}, year={2021}, url={}, note=[Accessed: 20-March-2023 ]}