CarlsonRC
✖
CarlsonRC
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
open allclose allBasic Examples (3)Summary of the most common use cases

https://wolfram.com/xid/0yv28ihk9dad-gp7yrq


https://wolfram.com/xid/0yv28ihk9dad-q2d8w

CarlsonRC is related to the special case of
for
:

https://wolfram.com/xid/0yv28ihk9dad-clyvt1


https://wolfram.com/xid/0yv28ihk9dad-rpyzk

Scope (13)Survey of the scope of standard use cases
Numerical Evaluation (6)

https://wolfram.com/xid/0yv28ihk9dad-d9ouir


https://wolfram.com/xid/0yv28ihk9dad-7ua9z

Precision of the output tracks the precision of the input:

https://wolfram.com/xid/0yv28ihk9dad-de8qhd


https://wolfram.com/xid/0yv28ihk9dad-fnvnoh

Evaluate for complex arguments:

https://wolfram.com/xid/0yv28ihk9dad-hl6r9f

Evaluate efficiently at high precision:

https://wolfram.com/xid/0yv28ihk9dad-j7u9wi


https://wolfram.com/xid/0yv28ihk9dad-b8un9d

CarlsonRC threads elementwise over lists:

https://wolfram.com/xid/0yv28ihk9dad-cltzmb

CarlsonRC can be used with Interval and CenteredInterval objects:

https://wolfram.com/xid/0yv28ihk9dad-bnbpn


https://wolfram.com/xid/0yv28ihk9dad-licshd

Specific Values (2)
Simple exact values are generated automatically:

https://wolfram.com/xid/0yv28ihk9dad-ikd9gc


https://wolfram.com/xid/0yv28ihk9dad-ejorb3


https://wolfram.com/xid/0yv28ihk9dad-bzgkjz


https://wolfram.com/xid/0yv28ihk9dad-e82e4d

Use FunctionExpand to convert CarlsonRC to elementary functions:

https://wolfram.com/xid/0yv28ihk9dad-0x0hy


https://wolfram.com/xid/0yv28ihk9dad-fym5lj

Differentiation and Integration (2)
Derivative of with respect to
:

https://wolfram.com/xid/0yv28ihk9dad-gt3bh3

Derivative of with respect to
:

https://wolfram.com/xid/0yv28ihk9dad-g1agv2

Indefinite integral of with respect to
:

https://wolfram.com/xid/0yv28ihk9dad-dlou9u

Indefinite integral of with respect to
:

https://wolfram.com/xid/0yv28ihk9dad-eksbcy

Function Representations (1)
Applications (3)Sample problems that can be solved with this function
Use CarlsonRC to provide upper and lower bounds for CarlsonRF[x,y,z]:

https://wolfram.com/xid/0yv28ihk9dad-cjwsbr


https://wolfram.com/xid/0yv28ihk9dad-ga9k1m

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

https://wolfram.com/xid/0yv28ihk9dad-gom85s


https://wolfram.com/xid/0yv28ihk9dad-celfct


https://wolfram.com/xid/0yv28ihk9dad-lrc4uw

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

https://wolfram.com/xid/0yv28ihk9dad-ee6zuy

Properties & Relations (3)Properties of the function, and connections to other functions
For ,
can be expressed in terms of ArcCos:

https://wolfram.com/xid/0yv28ihk9dad-izws6k


https://wolfram.com/xid/0yv28ihk9dad-jly75l

is interpreted as a Cauchy principal value if
lies on the negative real axis:

https://wolfram.com/xid/0yv28ihk9dad-bcm5vd


https://wolfram.com/xid/0yv28ihk9dad-h1h6az


https://wolfram.com/xid/0yv28ihk9dad-g4abx9

Compare with the equivalent expression in terms of positive arguments:

https://wolfram.com/xid/0yv28ihk9dad-b1xi70

Use FunctionExpand to express CarlsonRC in terms of simpler functions:

https://wolfram.com/xid/0yv28ihk9dad-jjg66z


https://wolfram.com/xid/0yv28ihk9dad-bbnen1

Possible Issues (1)Common pitfalls and unexpected behavior
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:

https://wolfram.com/xid/0yv28ihk9dad-ganhsh


https://wolfram.com/xid/0yv28ihk9dad-btd4h3

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).
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.
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
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: 10-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: 10-July-2025
]}