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CarlsonRC
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CarlsonRC

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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

open allclose all

Basic Examples  (3)Summary of the most common use cases

Evaluate numerically:

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-gp7yrq
Out[1]=1

Plot the function:

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-q2d8w
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-clyvt1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv28ihk9dad-rpyzk
Out[2]=2

Scope  (13)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

In[2]:=2
✖
https://wolfram.com/xid/0yv28ihk9dad-d9ouir
Out[2]=2

Evaluate at high precision:

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-7ua9z
Out[1]=1

Precision of the output tracks the precision of the input:

In[2]:=2
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https://wolfram.com/xid/0yv28ihk9dad-de8qhd
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0yv28ihk9dad-fnvnoh
Out[3]=3

Evaluate for complex arguments:

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-hl6r9f
Out[1]=1

Evaluate efficiently at high precision:

In[1]:=1
✖
https://wolfram.com/xid/0yv28ihk9dad-j7u9wi
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0yv28ihk9dad-b8un9d
Out[2]=2

CarlsonRC threads elementwise over lists:

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-cltzmb
Out[1]=1

CarlsonRC can be used with Interval and CenteredInterval objects:

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-bnbpn
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0yv28ihk9dad-licshd
Out[2]=2

Specific Values  (2)

Simple exact values are generated automatically:

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-ikd9gc
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv28ihk9dad-ejorb3
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0yv28ihk9dad-bzgkjz
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0yv28ihk9dad-e82e4d
Out[4]=4

Use FunctionExpand to convert CarlsonRC to elementary functions:

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-0x0hy
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0yv28ihk9dad-fym5lj
Out[2]=2

Differentiation and Integration  (2)

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

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-gt3bh3
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0yv28ihk9dad-g1agv2
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-dlou9u
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0yv28ihk9dad-eksbcy
Out[2]=2

Function Representations  (1)

TraditionalForm formatting:

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-k9gpun

Function Identities and Simplifications  (2)

CarlsonRC satisfies the Euler–Poisson partial differential equation:

In[1]:=1
✖
https://wolfram.com/xid/0yv28ihk9dad-fpunj
Out[1]=1

CarlsonRC satisfies Euler's homogeneity relation:

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-zxe9v
Out[1]=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]:

In[5]:=5
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https://wolfram.com/xid/0yv28ihk9dad-cjwsbr
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0yv28ihk9dad-ga9k1m
Out[6]=6

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

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-gom85s
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0yv28ihk9dad-celfct
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0yv28ihk9dad-lrc4uw
Out[3]=3

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

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-ee6zuy
Out[1]=1

Properties & Relations  (3)Properties of the function, and connections to other functions

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

In[1]:=1
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https://wolfram.com/xid/0yv28ihk9dad-izws6k
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv28ihk9dad-jly75l
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0yv28ihk9dad-bcm5vd
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv28ihk9dad-h1h6az
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0yv28ihk9dad-g4abx9
Out[3]=3

Compare with the equivalent expression in terms of positive arguments:

In[4]:=4
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https://wolfram.com/xid/0yv28ihk9dad-b1xi70
Out[4]=4

Use FunctionExpand to express CarlsonRC in terms of simpler functions:

In[1]:=1
✖
https://wolfram.com/xid/0yv28ihk9dad-jjg66z
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv28ihk9dad-bbnen1
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

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:

In[1]:=1
✖
https://wolfram.com/xid/0yv28ihk9dad-ganhsh
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0yv28ihk9dad-btd4h3
Out[2]=2
Wolfram Research (2021), CarlsonRC, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRC.html (updated 2023).
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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.

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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 ]}

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@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 ]}

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@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 ]}