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CarlsonRM[x,y,ρ]

gives Carlson's elliptic integral TemplateBox[{x, y, rho}, CarlsonRM].

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Differentiation and Integration  
Function Representations  
Function Identities and Simplifications  
Applications  
Properties & Relations  
See Also
Related Guides
History
Cite this Page

CarlsonRM

CarlsonRM[x,y,ρ]

gives Carlson's elliptic integral TemplateBox[{x, y, rho}, CarlsonRM].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For non-negative arguments, TemplateBox[{x, y, rho}, CarlsonRM]=2/piint_0^inftyt^(-1/2) (t+x)^(-1/2)(t+y)^(-1/2)(t+rho)^(-1)dt.
  • CarlsonRM[x,y,ρ] has a branch cut discontinuity at .
  • CarlsonRM[x,y,ρ] is understood as a Cauchy principal value integral for ρ<0.
  • For certain arguments, CarlsonRM automatically evaluates to exact values.
  • CarlsonRM can be evaluated to arbitrary precision.
  • CarlsonRM automatically threads over lists.

Examples

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Basic Examples  (3)

Evaluate numerically:

Plot the function:

CarlsonRM is related to the complete Legendre elliptic integral of the third kind:

Scope  (11)

Numerical Evaluation  (5)

Evaluate numerically:

Evaluate numerically to high precision:

Precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate efficiently at high precision:

CarlsonRM threads elementwise over lists:

Specific Values  (1)

Simple exact values are generated automatically:

Differentiation and Integration  (2)

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

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

Indefinite integral of TemplateBox[{x, y, rho}, CarlsonRM] with respect to :

Function Representations  (1)

TraditionalForm formatting:

Function Identities and Simplifications  (2)

CarlsonRM satisfies the Euler–Poisson partial differential equation:

CarlsonRM satisfies Euler's homogeneity relation:

Applications  (2)

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 expressed in terms of Carlson integrals:

Compare with the result of Volume:

Properties & Relations  (1)

CarlsonRM is symmetric with respect to its first two arguments:

Wolfram Research (2021), CarlsonRM, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRM.html.

Text

Wolfram Research (2021), CarlsonRM, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRM.html.

CMS

Wolfram Language. 2021. "CarlsonRM." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CarlsonRM.html.

APA

Wolfram Language. (2021). CarlsonRM. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CarlsonRM.html

BibTeX

@misc{reference.wolfram_2025_carlsonrm, author="Wolfram Research", title="{CarlsonRM}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRM.html}", note=[Accessed: 31-October-2025]}

BibLaTeX

@online{reference.wolfram_2025_carlsonrm, organization={Wolfram Research}, title={CarlsonRM}, year={2021}, url={https://reference.wolfram.com/language/ref/CarlsonRM.html}, note=[Accessed: 31-October-2025]}