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


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For non-negative arguments, TemplateBox[{x, y}, CarlsonRE]⩵1/piint_0^infty(t+x)^(-1/2)(t+y)^(-1/2)sqrt(t)( x/(t+x)+y/(t+y))dt.
  • 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.


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Basic Examples  (3)

Evaluate numerically:

Plot the function:

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

Scope  (11)

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:

Function Identities and Simplifications  (3)

CarlsonRE satisfies the EulerPoisson partial differential equation:

CarlsonRE satisfies Euler's homogeneity relation:

A partial differential equation satisfied by CarlsonRE:

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 with the 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:

Properties & Relations  (2)

CarlsonRE is invariant under a permutation of its arguments:

CarlsonRE and CarlsonRK satisfy Legendre's relation:

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_2024_carlsonre, author="Wolfram Research", title="{CarlsonRE}", year="2021", howpublished="\url{}", note=[Accessed: 22-May-2024 ]}


@online{reference.wolfram_2024_carlsonre, organization={Wolfram Research}, title={CarlsonRE}, year={2021}, url={}, note=[Accessed: 22-May-2024 ]}