CarlsonRC

CarlsonRC[x,y]

gives the Carlson's elliptic integral .

Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• For and , .
• 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 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 with respect to :

Derivative of with respect to :

Indefinite integral of with respect to :

Indefinite integral of with respect to :

Function Identities and Simplifications(2)

CarlsonRC satisfies the EulerPoisson 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 , can be expressed in terms of ArcCos:

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, ; 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_2024_carlsonrc, author="Wolfram Research", title="{CarlsonRC}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRC.html}", note=[Accessed: 19-September-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_carlsonrc, organization={Wolfram Research}, title={CarlsonRC}, year={2023}, url={https://reference.wolfram.com/language/ref/CarlsonRC.html}, note=[Accessed: 19-September-2024 ]}