gives the Carlson's elliptic integral .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For non-negative arguments, .
  • CarlsonRE[x,y] has a branch cut discontinuity at .
  • For certain arguments, CarlsonRE automatically evaluates to exact values.
  • CarlsonRE can be evaluated to arbitrary numerical precision.
  • CarlsonRE automatically threads over lists.


open allclose all

Basic Examples  (3)

Evaluate numerically:

Plot the function:

CarlsonRE is related to Legendre's complete elliptic integral of the second kind:

Scope  (8)

Numerical Evaluation  (5)

Evaluate numerically:

Evaluate to high precision:

Precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate efficiently at high precision:

CarlsonRE threads elementwise over lists:

Specific Values  (1)

Simple exact values are generated automatically:

Derivatives  (1)


Functional Representation  (1)

TraditionalForm formatting:

Applications  (3)

Total arc length of an ellipse:

Compare with the result of ArcLength:

Expectation value of the square root of a quadratic form, relative to a normal distribution:

Compare to closed-form result in terms of CarlsonRE:

Visualize the intersection of a cylinder and a ball:

Volume of cylinder-ball intersection expressed in terms of Carlson integrals:

Compare with the result of Volume:

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


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


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


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


@misc{reference.wolfram_2023_carlsonre, author="Wolfram Research", title="{CarlsonRE}", year="2021", howpublished="\url{}", note=[Accessed: 07-December-2023 ]}


@online{reference.wolfram_2023_carlsonre, organization={Wolfram Research}, title={CarlsonRE}, year={2021}, url={}, note=[Accessed: 07-December-2023 ]}