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CarlsonRM[x,y,ρ]

给出 Carlson 椭圆积分 TemplateBox[{x, y, rho}, CarlsonRM]

更多信息
Details and Options Details and Options
范例  
基本范例  
范围  
数值运算  
特定值  
导数和积分  
函数表示  
函数恒等和化简  
应用  
属性和关系  
参见
相关指南
历史
按以下格式引用：

CarlsonRM

CarlsonRM[x,y,ρ]

给出 Carlson 椭圆积分 TemplateBox[{x, y, rho}, CarlsonRM]

更多信息

  • 数学函数，适用于符号和数字运算.
  • 对于非负参数，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] 在 处有分支切割.
  • 对于 ρ<0，CarlsonRM[x,y,ρ] 可以理解为柯西主值积分.
  • 对于某些参数，CarlsonRM 会自动求出精确值.
  • CarlsonRM 可以运算至任意精度.
  • CarlsonRM 自动遍历列表.

范例

打开所有单元 关闭所有单元

基本范例  (3)

数值运算：

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

绘制函数图形：

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

CarlsonRM 与第三类完整勒让德椭圆积分有关：

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}

范围  (11)

数值运算  (5)

数值运算：

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

高精度运算：

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

输出的精度与输入的精度一致：

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

对复参数进行运算：

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

高精度高效运算：

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

CarlsonRM 按元素遍历列表：

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]}]

特定值  (1)

简单的精确结果是自动生成的：

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]

导数和积分  (2)

TemplateBox[{x, y, rho}, CarlsonRM] 关于 的导数：

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

TemplateBox[{x, y, rho}, CarlsonRM] 关于 的导数：

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

TemplateBox[{x, y, rho}, CarlsonRM] 关于 的不定积分：

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

函数表示  (1)

TraditionalForm 格式：

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

函数恒等和化简  (2)

CarlsonRM 满足欧拉-泊松偏微分方程：

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 满足欧拉齐次关系：

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

应用  (2)

可视化圆盘相对的立体角：

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.}]]

运算立体角：

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]]

与 NIntegrate 的结果比较：

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}]]

可视化圆柱体和球体的交集：

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"]}

圆柱-球交集的体积：

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]

与 Volume 的结果比较：

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

属性和关系  (1)

CarlsonRM 就其前两个参数而言是对称函数：

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

文本

Wolfram Research (2021)，CarlsonRM，Wolfram 语言函数，https://reference.wolfram.com/language/ref/CarlsonRM.html.

CMS

Wolfram 语言. 2021. "CarlsonRM." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/CarlsonRM.html.

APA

Wolfram 语言. (2021). CarlsonRM. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/CarlsonRM.html 年

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: 09-September-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: 09-September-2026]}