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


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For non-negative arguments, TemplateBox[{x, y}, CarlsonRK]⩵1/piint_0^inftyt^(-1/2)(t+x)^(-1/2)(t+y)^(-1/2)dt.
  • CarlsonRK[x,y] has a branch cut discontinuity at .
  • For certain arguments, CarlsonRK automatically evaluates to exact values.
  • CarlsonRK can be evaluated to arbitrary precision.
  • CarlsonRK automatically threads over lists.


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Basic Examples  (3)

Evaluate numerically:

Plot the function:

CarlsonRK is related to the complete elliptic integral of the first kind EllipticK:

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:

CarlsonRK threads elementwise over lists:

Specific Values  (1)

Simple exact values are generated automatically:

Derivatives  (1)

Derivative with respect to :

Derivative with respect to :

Functional Representation  (1)

TraditionalForm formatting:

Function Identities and Simplifications  (3)

CarlsonRK satisfies the EulerPoisson partial differential equation:

CarlsonRK satisfies Euler's homogeneity relation:

A partial differential equation satisfied by CarlsonRK:

Applications  (5)

Total arc length of a lemniscate of Bernoulli:

Compare with the result of ArcLength:

Evaluate an elliptic singular value:

Expectation value of the reciprocal square root of a quadratic form over a normal distribution:

Compare with the closed-form result in terms of CarlsonRK:

Visualize the solid angle subtended by a circular disk:

Evaluate the solid angle:

Compare with the result of NIntegrate:

Visualize the intersection of a cylinder and a ball:

Volume of cylinder-ball intersection:

Compare with the result of Volume:

Properties & Relations  (3)

CarlsonRK is invariant under a permutation of its arguments:

CarlsonRK and CarlsonRE satisfy Legendre's relation:

CarlsonRK is related to ArithmeticGeometricMean:

Neat Examples  (1)

Probability that a random walker in a 3D cubic lattice returns to the origin:

Carry out a modeling run of 1000 walks and count how many return to the origin:

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


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


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


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


@misc{reference.wolfram_2024_carlsonrk, author="Wolfram Research", title="{CarlsonRK}", year="2021", howpublished="\url{}", note=[Accessed: 19-June-2024 ]}


@online{reference.wolfram_2024_carlsonrk, organization={Wolfram Research}, title={CarlsonRK}, year={2021}, url={}, note=[Accessed: 19-June-2024 ]}