CarlsonRM [x,y,ρ]
gives Carlson's elliptic integral TemplateBox[{x, y, rho}, CarlsonRM].
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
open all close allBasic 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:
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:
See Also
Related Guides
History
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: 11-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_carlsonrm, organization={Wolfram Research}, title={CarlsonRM}, year={2021}, url={https://reference.wolfram.com/language/ref/CarlsonRM.html}, note=[Accessed: 11-January-2026]}