CarlsonRE [x,y]
gives the Carlson's elliptic integral TemplateBox[{x, y}, CarlsonRE].
CarlsonRE
CarlsonRE [x,y]
gives the Carlson's elliptic integral TemplateBox[{x, y}, CarlsonRE].
Details
- 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.
Examples
open all close allBasic Examples (3)
Evaluate numerically:
Plot the function:
CarlsonRE is related to Legendre's complete elliptic integral of the second kind:
Scope (12)
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:
Differentiation and Integration (2)
Derivative with respect to :
Derivative with respect to :
Indefinite integral with respect to :
Indefinite integral with respect to :
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 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 :
Related Guides
History
Text
Wolfram Research (2021), CarlsonRE, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRE.html.
CMS
Wolfram Language. 2021. "CarlsonRE." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CarlsonRE.html.
APA
Wolfram Language. (2021). CarlsonRE. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CarlsonRE.html
BibTeX
@misc{reference.wolfram_2025_carlsonre, author="Wolfram Research", title="{CarlsonRE}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRE.html}", note=[Accessed: 09-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_carlsonre, organization={Wolfram Research}, title={CarlsonRE}, year={2021}, url={https://reference.wolfram.com/language/ref/CarlsonRE.html}, note=[Accessed: 09-January-2026]}