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CarlsonRC[x,y]

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

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Differentiation and Integration  
Function Representations  
Function Identities and Simplifications  
Applications  
Properties & Relations  
Possible Issues  
See Also
Related Guides
History
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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:

Wolfram Language code: CarlsonRC[4., 5.]

Plot the function:

Wolfram Language code: Plot3D[CarlsonRC[x, y], {x, 0, 3}, {y, 0, 3}]

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

Wolfram Language code: Sin[ϕ]CarlsonRC[Cos[ϕ]^2, 1] /. ϕ -> 0.3
Wolfram Language code: EllipticF[0.3, 0]

Scope  (13)

Numerical Evaluation  (6)

Evaluate numerically:

Wolfram Language code: CarlsonRC[5, 4.]

Evaluate at high precision:

Wolfram Language code: N[CarlsonRC[Sqrt[3], -2], 50]

Precision of the output tracks the precision of the input:

Wolfram Language code: CarlsonRC[3, 1.234567890123456789012345]
Wolfram Language code: CarlsonRC[3, 1.2345678901234567890123456789012345]

Evaluate for complex arguments:

Wolfram Language code: CarlsonRC[Exp[I Pi / 7.], Exp[I Pi / 3.]]

Evaluate efficiently at high precision:

Wolfram Language code: Timing[CarlsonRC[3, 5`500]]
Wolfram Language code: Timing[Precision[CarlsonRC[3, 5`100000]]]

CarlsonRC threads elementwise over lists:

Wolfram Language code: CarlsonRC[{1, 2, 3, 4, 5}, 3.]

CarlsonRC can be used with Interval and CenteredInterval objects:

Wolfram Language code: CarlsonRC[Interval[{1.23, 1.24}], Interval[{2.34, 2.35}]]
Wolfram Language code: CarlsonRC[CenteredInterval[3 / 2, 1 / 100], CenteredInterval[5 / 4, 1 / 100]]

Specific Values  (2)

Simple exact values are generated automatically:

Wolfram Language code: CarlsonRC[7, 7]
Wolfram Language code: CarlsonRC[0, 4]
Wolfram Language code: CarlsonRC[-Pi, -Pi]
Wolfram Language code: CarlsonRC[x, 0]

Use FunctionExpand to convert CarlsonRC to elementary functions:

Wolfram Language code: CarlsonRC[4, 2]//FunctionExpand
Wolfram Language code: CarlsonRC[3 - 2I, 1 + 5I]//FunctionExpand

Differentiation and Integration  (2)

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

Wolfram Language code: D[CarlsonRC[x, y], x]

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

Wolfram Language code: D[CarlsonRC[x, y], y]

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

Wolfram Language code: Integrate[CarlsonRC[x, y], x]

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

Wolfram Language code: Integrate[CarlsonRC[x, y], y]

Function Representations  (1)

TraditionalForm formatting:

Wolfram Language code: CarlsonRC[x, y]//TraditionalForm

Function Identities and Simplifications  (2)

CarlsonRC satisfies the Euler–Poisson partial differential equation:

Wolfram Language code: (x - y)Subscript[∂, x⁣y]CarlsonRC[x, y] - Subscript[∂, x]CarlsonRC[x, y] + (1/2)Subscript[∂, y]CarlsonRC[x, y] == 0//FullSimplify

CarlsonRC satisfies Euler's homogeneity relation:

Wolfram Language code: xSubscript[∂, x]CarlsonRC[x, y] + ySubscript[∂, y]CarlsonRC[x, y] == -(1/2)CarlsonRC[x, y]//FullSimplify

Applications  (3)

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

Wolfram Language code: With[{x = 3, y = 7, z = 9}, CarlsonRC[x, (y + z/2)] ≤ CarlsonRF[x, y, z] ≤ CarlsonRC[x, Sqrt[y z]]]
Wolfram Language code: With[{x = 3, y = 7., z = 9}, {CarlsonRC[x, (y + z/2)], CarlsonRF[x, y, z], CarlsonRC[x, Sqrt[y z]]}]

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

Wolfram Language code: EllipticPi[n, ϕ, m] + EllipticPi[m / n, ϕ, m] == EllipticF[ϕ, m] + Csc[ϕ]CarlsonRC[(Csc[ϕ]^2 - 1)(Csc[ϕ]^2 - m), (Csc[ϕ]^2 - n)(Csc[ϕ]^2 - m / n)] /. {{n -> 1 / 3, ϕ -> Pi / 3, m -> 1 / 4.}, {n -> 2, ϕ -> Pi / 3, m -> 1 / 4.}, {n -> -1, ϕ -> Pi / 3, m -> 1 / 4.}}
Wolfram Language code: (m - n)EllipticPi[n, ϕ, m] + (m - (m - n/1 - n))EllipticPi[(m - n/1 - n), ϕ, m] == m EllipticF[ϕ, m] - n (m - n/1 - n)Cot[ϕ]CarlsonRC[Csc[ϕ]^2(Csc[ϕ]^2 - m), (Csc[ϕ]^2 - n)(Csc[ϕ]^2 - (m - n/1 - n))] /. {{n -> 1 / 3, ϕ -> Pi / 3, m -> 1 / 4.}, {n -> 2, ϕ -> Pi / 3, m -> 1 / 4.}, {n -> -1, ϕ -> Pi / 3, m -> 1 / 4.}}
Wolfram Language code: (1 - n)EllipticPi[n, ϕ, m] + (1 - (m (1 - n)/m - n))EllipticPi[(m (1 - n)/m - n), ϕ, m] == EllipticF[ϕ, m] + (1 - n - (m (1 - n)/m - n))Sqrt[Csc[ϕ]^2 - m]CarlsonRC[Cot[ϕ]^2 Csc[ϕ]^2, (Csc[ϕ]^2 - n)(Csc[ϕ]^2 - (m (1 - n)/m - n))] /. {{n -> 1 / 3, ϕ -> Pi / 3, m -> 1 / 4.}, {n -> 2, ϕ -> Pi / 3, m -> 1 / 4.}, {n -> -1, ϕ -> Pi / 3, m -> 1 / 4.}}

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

Wolfram Language code: With[{q = x + ((y - x)(z - x)/p - x)}, (p - x)CarlsonRJ[x, y, z, p] + (q - x)CarlsonRJ[x, y, z, q] == 3CarlsonRF[x, y, z] - 3CarlsonRC[(y z/x), (p q/x)]] /. {{x -> 2, y -> 3, z -> 5, p -> 7.}, {x -> 2, y -> 3, z -> 5, p -> 1.}, {x -> 2, y -> 3, z -> 5, p -> -4.}}

Properties & Relations  (3)

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

Wolfram Language code: With[{x = 4, y = 7}, CarlsonRC[x, y] == (1/Sqrt[y - x])ArcCos[Sqrt[(x/y)]]]
Wolfram Language code: N[%, 100]

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

Wolfram Language code: CarlsonRC[3, -7`]
Wolfram Language code: {CarlsonRC[3, -7` + I $MachineEpsilon], CarlsonRC[3, -7` - I $MachineEpsilon]}
Wolfram Language code: Mean[%]

Compare with the equivalent expression in terms of positive arguments:

Wolfram Language code: {CarlsonRC[x, z], Sqrt[(x/x - z)]CarlsonRC[x - z, -z]} /. {x -> 3., z -> -7.}

Use FunctionExpand to express CarlsonRC in terms of simpler functions:

Wolfram Language code: CarlsonRC[4, 7]//FunctionExpand
Wolfram Language code: CarlsonRC[3, -7]//FunctionExpand

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 Language code: CarlsonRF[x, z, z]
Wolfram Language code: % /. z -> Pi
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_2026_carlsonrc, author="Wolfram Research", title="{CarlsonRC}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRC.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_carlsonrc, organization={Wolfram Research}, title={CarlsonRC}, year={2023}, url={https://reference.wolfram.com/language/ref/CarlsonRC.html}, note=[Accessed: 12-June-2026]}