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CarlsonRM[x,y,ρ]

gives Carlson's elliptic integral TemplateBox[{x, y, rho}, CarlsonRM].

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  
See Also
Related Guides
History
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CarlsonRM

CarlsonRM[x,y,ρ]

gives Carlson's elliptic integral TemplateBox[{x, y, rho}, CarlsonRM].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For non-negative arguments, TemplateBox[{x, y, rho}, CarlsonRM]=2/piint_0^inftyt^(-1/2) (t+x)^(-1/2)(t+y)^(-1/2)(t+rho)^(-1)dt.
  • CarlsonRM[x,y,ρ] has a branch cut discontinuity at .
  • CarlsonRM[x,y,ρ] is understood as a Cauchy principal value integral for ρ<0.
  • For certain arguments, CarlsonRM automatically evaluates to exact values.
  • CarlsonRM can be evaluated to arbitrary precision.
  • CarlsonRM automatically threads over lists.

Examples

open all close all

Basic Examples  (3)

Evaluate numerically:

Wolfram Language code: CarlsonRM[3., 5., 1.]

Plot the function:

Wolfram Language code: Plot[{CarlsonRM[t, 5, 7], CarlsonRM[3, t, 7], CarlsonRM[3, 5, t]}, {t, -10, 10}, PlotLegends -> "Expressions"]

CarlsonRM is related to the complete Legendre elliptic integral of the third kind:

Wolfram Language code: n Pi / 4 CarlsonRM[1 - m, 1, 1 - n] /. {m -> 0.4, n -> -4}
Wolfram Language code: EllipticPi[n, m] - EllipticK[m] /. {m -> 0.4, n -> -4}

Scope  (11)

Numerical Evaluation  (5)

Evaluate numerically:

Wolfram Language code: N[CarlsonRM[Sqrt[3], 4, 2]]

Evaluate numerically to high precision:

Wolfram Language code: N[CarlsonRM[5, 1, 2], 50]

Precision of the output tracks the precision of the input:

Wolfram Language code: CarlsonRM[5, 1, 2.34567890123456789012345]
Wolfram Language code: CarlsonRM[5, 1, 2.345678901234567890123456789012345]

Evaluate for complex arguments:

Wolfram Language code: N[CarlsonRM[-3 + I, -2, 7 + I]]

Evaluate efficiently at high precision:

Wolfram Language code: Timing[CarlsonRM[3, 5, 11`500]]
Wolfram Language code: Timing[Precision[CarlsonRM[3, 5, 11`100000]]]

CarlsonRM threads elementwise over lists:

Wolfram Language code: CarlsonRM[{Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}, {Subscript[y, 1], Subscript[y, 2], Subscript[y, 3]}, {Subscript[ρ, 1], Subscript[ρ, 2], Subscript[ρ, 3]}]

Specific Values  (1)

Simple exact values are generated automatically:

Wolfram Language code: CarlsonRM[x, x, x]
Wolfram Language code: CarlsonRM[x, x, ρ]
Wolfram Language code: CarlsonRM[x, y, 0]
Wolfram Language code: CarlsonRM[x, 0, ρ]

Differentiation and Integration  (2)

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

Wolfram Language code: D[CarlsonRM[x, y, ρ], x]

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

Wolfram Language code: D[CarlsonRM[x, y, ρ], ρ]

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

Wolfram Language code: Integrate[CarlsonRM[x, y, ρ], x]

Function Representations  (1)

TraditionalForm formatting:

Wolfram Language code: CarlsonRM[x, y, z]//TraditionalForm

Function Identities and Simplifications  (2)

CarlsonRM satisfies the Euler–Poisson partial differential equation:

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

CarlsonRM satisfies Euler's homogeneity relation:

Wolfram Language code: xSubscript[∂, x]CarlsonRM[x, y, ρ] + ySubscript[∂, y]CarlsonRM[x, y, ρ] + ρSubscript[∂, ρ]CarlsonRM[x, y, ρ] == -(3/2)CarlsonRM[x, y, ρ]//FunctionExpand//FullSimplify

Applications  (2)

Visualize the solid angle subtended by a circular disk:

Wolfram Language code: With[{L = 2, r0 = 2 / 5, rm = 1}, Graphics3D[{EdgeForm[], Polygon[PadRight[N@CirclePoints[rm, 24], {Automatic, 3}]], {Dashed, Line[{{r0, 0, 0}, {r0, 0, L}}]}, Sphere[{r0, 0, L}, rm / 20]}, Boxed -> False, ViewPoint -> {-2.4, -1.3, 2.}]]

Evaluate the solid angle:

Wolfram Language code: With[{L = 2, r0 = 2 / 5, rm = 1}, N[2π Boole[r0 < rm] - (2π L rm/rm - r0)(CarlsonRK[L^2 + (rm - r0)^2, L^2 + (rm + r0)^2] - (r0(L^2 + (rm - r0)^2)/(rm^2 - r0^2)^2) CarlsonRM[(L^2 + (rm + r0)^2/(rm + r0)^2), (L^2 + (rm - r0)^2/(rm + r0)^2), (L^2 + (rm - r0)^2/(rm - r0)^2)]), 20]]

Compare with the result of NIntegrate:

Wolfram Language code: With[{L = 2, r0 = 2 / 5, rm = 1}, L NIntegrate[(r/(r^2 - 2r0 r Cos[θ] + r0^2 + L^2)^3 / 2), {r, 0, rm}, {θ, 0, 2π}]]

Visualize the intersection of a cylinder and a ball:

Wolfram Language code: R = 1 / 6;r = 5 / 9;b = 5 / 11; cyl = Cylinder[{{0, 0, 0}, {0, 0, 2r}}, R]; ball = Ball[{b, 0, r}, r];
Wolfram Language code: Show@{ Graphics3D[{Opacity[1 / 2], cyl, ball}], Region[RegionIntersection[cyl, ball], PlotTheme -> "Web"]}

Volume of cylinder-ball intersection expressed in terms of Carlson integrals:

Wolfram Language code: N[(2π/9 )(2R(3 R (2 r^2 - R^2) - b (b^2 - 4 r^2 + 5 b R + 7 R^2) + (3 r^4/b - R))CarlsonRK[(b - r + R) (b + r + R), 4b R] + (b^2 - 4 r^2 + 7 R^2)CarlsonRE[(b - r + R) (b + r + R), 4b R] + 6b R r^4(b + r - R) (b + R) (b - r - R)CarlsonRM[(b - R)^2 (b - r + R) (b + r + R), 4 b R(b - R)^2, 4 b R r^2]), 20]

Compare with the result of Volume:

Wolfram Language code: Volume[RegionIntersection[cyl, ball], WorkingPrecision -> 20]

Properties & Relations  (1)

CarlsonRM is symmetric with respect to its first two arguments:

Wolfram Language code: CarlsonRM[x, y, ρ] == CarlsonRM[y, x, ρ]
Wolfram Research (2021), CarlsonRM, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRM.html.

Text

Wolfram Research (2021), CarlsonRM, Wolfram Language function, https://reference.wolfram.com/language/ref/CarlsonRM.html.

CMS

Wolfram Language. 2021. "CarlsonRM." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CarlsonRM.html.

APA

Wolfram Language. (2021). CarlsonRM. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CarlsonRM.html

BibTeX

@misc{reference.wolfram_2026_carlsonrm, author="Wolfram Research", title="{CarlsonRM}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRM.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_carlsonrm, organization={Wolfram Research}, title={CarlsonRM}, year={2021}, url={https://reference.wolfram.com/language/ref/CarlsonRM.html}, note=[Accessed: 12-June-2026]}