CarlsonRD
✖
CarlsonRD
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- For non-negative arguments,
.
- CarlsonRD[x,y,z] has a branch cut discontinuity at
.
- For certain arguments, CarlsonRD automatically evaluates to exact values.
- CarlsonRD can be evaluated to arbitrary precision.
- CarlsonRD automatically threads over lists.
- CarlsonRD 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/0bnhv6qov8gan-fdqy05

Plot CarlsonRD:

https://wolfram.com/xid/0bnhv6qov8gan-e75yio

CarlsonRD is related to a combination of Legendre elliptic integrals restricted to :

https://wolfram.com/xid/0bnhv6qov8gan-drf4tj


https://wolfram.com/xid/0bnhv6qov8gan-cxvwsw

Scope (15)Survey of the scope of standard use cases
Numerical Evaluation (6)
Evaluate CarlsonRD numerically:

https://wolfram.com/xid/0bnhv6qov8gan-95hog


https://wolfram.com/xid/0bnhv6qov8gan-mhyk9j

The precision of the output tracks the precision of the input:

https://wolfram.com/xid/0bnhv6qov8gan-gdzwv


https://wolfram.com/xid/0bnhv6qov8gan-d5jot0

Evaluate for complex arguments:

https://wolfram.com/xid/0bnhv6qov8gan-cljtsd

Evaluate efficiently at high precision:

https://wolfram.com/xid/0bnhv6qov8gan-os5se


https://wolfram.com/xid/0bnhv6qov8gan-bjtg0

CarlsonRD threads elementwise over lists:

https://wolfram.com/xid/0bnhv6qov8gan-d20a4j

CarlsonRD can be used with Interval and CenteredInterval objects:

https://wolfram.com/xid/0bnhv6qov8gan-eql2zd


https://wolfram.com/xid/0bnhv6qov8gan-kh3mzr

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

https://wolfram.com/xid/0bnhv6qov8gan-i3elhq


https://wolfram.com/xid/0bnhv6qov8gan-ow5udj


https://wolfram.com/xid/0bnhv6qov8gan-c392ek

When one argument of CarlsonRD is zero, CarlsonRD can be expressed in terms of the complete elliptic integrals CarlsonRE and CarlsonRK:

https://wolfram.com/xid/0bnhv6qov8gan-fpi93j

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

https://wolfram.com/xid/0bnhv6qov8gan-ggq5f1

Derivative of with respect to
:

https://wolfram.com/xid/0bnhv6qov8gan-m3t8ei

Indefinite integral of with respect to
:

https://wolfram.com/xid/0bnhv6qov8gan-czjexe

Indefinite integral of with respect to
:

https://wolfram.com/xid/0bnhv6qov8gan-bjnpq5

Function Representations (1)
Function Identities and Simplifications (4)
An equation relating CarlsonRD, CarlsonRF and CarlsonRG:

https://wolfram.com/xid/0bnhv6qov8gan-ejtp1i

Some cyclic permutation identities for CarlsonRD:

https://wolfram.com/xid/0bnhv6qov8gan-go6zdh


https://wolfram.com/xid/0bnhv6qov8gan-gi4lsc


https://wolfram.com/xid/0bnhv6qov8gan-lngtb

CarlsonRD satisfies the Euler–Poisson partial differential equation:

https://wolfram.com/xid/0bnhv6qov8gan-fpunj


https://wolfram.com/xid/0bnhv6qov8gan-g1wbon

CarlsonRD satisfies Euler's homogeneity relation:

https://wolfram.com/xid/0bnhv6qov8gan-zxe9v

Applications (2)Sample problems that can be solved with this function
Distance along a meridian of the Earth:

https://wolfram.com/xid/0bnhv6qov8gan-hee91

Compare with the result of GeoDistance:

https://wolfram.com/xid/0bnhv6qov8gan-di7s2e

Parametrization of a Mylar balloon (two flat sheets of plastic sewn together at their circumference and then inflated):

https://wolfram.com/xid/0bnhv6qov8gan-e0kt4

Properties & Relations (1)Properties of the function, and connections to other functions
CarlsonRD is symmetric with respect to its first two arguments:

https://wolfram.com/xid/0bnhv6qov8gan-hqjxwc

Wolfram Research (2021), CarlsonRD, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRD.html (updated 2023).
Text
Wolfram Research (2021), CarlsonRD, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRD.html (updated 2023).
Wolfram Research (2021), CarlsonRD, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRD.html (updated 2023).
CMS
Wolfram Language. 2021. "CarlsonRD." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/CarlsonRD.html.
Wolfram Language. 2021. "CarlsonRD." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/CarlsonRD.html.
APA
Wolfram Language. (2021). CarlsonRD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CarlsonRD.html
Wolfram Language. (2021). CarlsonRD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CarlsonRD.html
BibTeX
@misc{reference.wolfram_2025_carlsonrd, author="Wolfram Research", title="{CarlsonRD}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRD.html}", note=[Accessed: 11-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_carlsonrd, organization={Wolfram Research}, title={CarlsonRD}, year={2023}, url={https://reference.wolfram.com/language/ref/CarlsonRD.html}, note=[Accessed: 11-July-2025
]}