CarlsonRC

CarlsonRC[x,y]

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

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For and , TemplateBox[{x, y}, CarlsonRC]=1/2int_0^infty(t+x)^(-1/2)(t+y)^(-1)dt.
  • CarlsonRC[x,y] has a branch cut discontinuity at .
  • CarlsonRC[x,y] is real valued for and , and is interpreted as a Cauchy principal value integral for .
  • For certain arguments, CarlsonRC automatically evaluates to exact values.
  • FunctionExpand can convert CarlsonRC to an expression in terms of elementary functions, whenever applicable.
  • CarlsonRC can be evaluated to arbitrary numerical precision.
  • CarlsonRC automatically threads over lists.
  • CarlsonRC can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (3)

Evaluate numerically:

Plot the function:

CarlsonRC is related to the special case of TemplateBox[{phi, m}, EllipticF] for :

Scope  (13)

Numerical Evaluation  (6)

Evaluate numerically:

Evaluate at high precision:

Precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate efficiently at high precision:

CarlsonRC threads elementwise over lists:

CarlsonRC can be used with Interval and CenteredInterval objects:

Specific Values  (2)

Simple exact values are generated automatically:

Use FunctionExpand to convert CarlsonRC to elementary functions:

Differentiation and Integration  (2)

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

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

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

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

Function Representations  (1)

TraditionalForm formatting:

Function Identities and Simplifications  (2)

CarlsonRC satisfies the Euler–Poisson partial differential equation:

CarlsonRC satisfies Euler's homogeneity relation:

Applications  (3)

Use CarlsonRC to provide upper and lower bounds for CarlsonRF[x,y,z]:

CarlsonRC is useful for compactly expressing the change of parameter relations for EllipticPi:

Use CarlsonRC to express the change of parameter relation for CarlsonRJ:

Properties & Relations  (3)

For , TemplateBox[{x, y}, CarlsonRC] can be expressed in terms of ArcCos:

TemplateBox[{x, y}, CarlsonRC] is interpreted as a Cauchy principal value if lies on the negative real axis:

Compare with the equivalent expression in terms of positive arguments:

Use FunctionExpand to express CarlsonRC in terms of simpler functions:

Possible Issues  (1)

Generically, TemplateBox[{x, z, z}, CarlsonRF]=TemplateBox[{x, z}, CarlsonRC]; however, due to differences in the analytic structures of the two functions, the evaluation is restricted to numeric values of that do not lie on the negative real axis:

Wolfram Research (2021), CarlsonRC, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRC.html (updated 2023).

Text

Wolfram Research (2021), CarlsonRC, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRC.html (updated 2023).

CMS

Wolfram Language. 2021. "CarlsonRC." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/CarlsonRC.html.

APA

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

BibTeX

@misc{reference.wolfram_2025_carlsonrc, author="Wolfram Research", title="{CarlsonRC}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRC.html}", note=[Accessed: 07-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_carlsonrc, organization={Wolfram Research}, title={CarlsonRC}, year={2023}, url={https://reference.wolfram.com/language/ref/CarlsonRC.html}, note=[Accessed: 07-July-2025 ]}