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CarlsonRK[x,y]

给出 Carlson 椭圆积分 TemplateBox[{x, y}, CarlsonRK].

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  
巧妙范例  
参见
相关指南
历史
按以下格式引用：

CarlsonRK

CarlsonRK[x,y]

给出 Carlson 椭圆积分 TemplateBox[{x, y}, CarlsonRK].

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}, CarlsonRK]=1/piint_0^inftyt^(-1/2)(t+x)^(-1/2)(t+y)^(-1/2)dt.
  • CarlsonRK[x,y] 在 处有分支切割.
  • 对于某些参数，CarlsonRK 会自动求出精确值.
  • CarlsonRK 可以运算至任意精度.
  • CarlsonRK 自动逐项作用于列表的各个元素.

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: CarlsonRK[2., 3.]

绘制函数图形：

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

CarlsonRK 与第一类完整椭圆积分 EllipticK 有关：

Wolfram Language code: π / 2 CarlsonRK[1, 1 - m] /. m -> 0.3
Wolfram Language code: EllipticK[m] /. m -> 0.3

范围  (12)

数值运算  (5)

数值运算：

Wolfram Language code: CarlsonRK[1.0, Sqrt[2] - 1]

高精度运算：

Wolfram Language code: N[CarlsonRK[5, 7], 50]

输出的精度与输入的精度一致：

Wolfram Language code: CarlsonRK[5, 7.67890123456789012345]
Wolfram Language code: CarlsonRK[5, 7.678901234567890123456789012345]

对复参数进行运算：

Wolfram Language code: CarlsonRK[1 + I, 1. - I]

高精度高效运算：

Wolfram Language code: Timing[CarlsonRK[11, 4`500]]
Wolfram Language code: Timing[Precision[CarlsonRK[11, 4`100000]]]

CarlsonRK 按元素遍历列表：

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

特定值  (1)

简单的精确结果是自动生成的：

Wolfram Language code: CarlsonRK[x, x]
Wolfram Language code: CarlsonRK[x, 0]

导数和积分  (2)

关于 的导数：

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

关于 的导数：

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

关于 的不定积分：

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

关于 的不定积分：

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

函数表示  (1)

TraditionalForm 格式：

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

函数恒等和化简  (3)

CarlsonRK 满足欧拉-泊松偏微分方程：

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

CarlsonRK 满足欧拉齐次关系：

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

CarlsonRK 满足的偏微分方程：

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

应用  (5)

伯努利双纽线的总弧长：

Wolfram Language code: N[2π CarlsonRK[1, 2], 25]

与 ArcLength 的结果比较：

Wolfram Language code: ArcLength[{(Cos[t]/1 + Sin[t]^2), (Sin[t]Cos[t]/1 + Sin[t]^2)}, {t, 0, 2π}, WorkingPrecision -> 25]

运算椭圆奇异值：

Wolfram Language code: N[{2π CarlsonRK[16, 8 + 3 Sqrt[7]], (Gamma[(1/7)]Gamma[(2/7)]Gamma[(4/7)]/47^(1/(4))π)}]

正态分布上二次形式的倒数平方根的期望值：

Wolfram Language code: MatrixForm[mat = HilbertMatrix[2]]
Wolfram Language code: NExpectation[(Sqrt[2]/Sqrt[{u, v}.mat.{u, v}]), {u, v}BinormalDistribution[0]]//Quiet

与 CarlsonRK 的闭形结果比较:

Wolfram Language code: N[Gamma[(1/2)](CarlsonRK@@Eigenvalues[mat]), 25]

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  (3)

CarlsonRK 在其参数排列置换后不变：

Wolfram Language code: CarlsonRK[x, y] == CarlsonRK[y, x]

CarlsonRK 和 CarlsonRE 满足 Legendre 关系式：

Wolfram Language code: CarlsonRE[y - x, y]CarlsonRK[x, y] + CarlsonRE[x, y]CarlsonRK[y - x, y] - y CarlsonRK[x, y]CarlsonRK[y - x, y] == (2/π) /. {{x -> 5`50, y -> 7`50}, {x -> 5`50, y -> 3`50}}

CarlsonRK 与 ArithmeticGeometricMean 相关：

Wolfram Language code: FullSimplify[CarlsonRK[x^2, y^2] == (1/ArithmeticGeometricMean[x, y]), x > 0 && y > 0]

巧妙范例  (1)

三维立方晶格中一个随机漫步者返回原点的概率：

Wolfram Language code: 1 - (6 + 2 Sqrt[3] + Sqrt[6]/18)CarlsonRK[1, 14 Sqrt[6] + 20 Sqrt[3] - 24 Sqrt[2] - 34]^-2 //N

执行一个 1000 次行走的建模，并计算返回原点的次数：

Wolfram Language code: BlockRandom[SeedRandom[11];Count[Table[walkerPosition = {0, 0, 0};steps = 0;While[steps++;walkerPosition += {{1, 0, 0}, {-1, 0, 0}, {0, 1, 0}, {0, -1, 0}, {0, 0, 1}, {0, 0, -1}}[[Random[Integer, {1, 6}]]]; steps < 100 && walkerPosition =!= {0, 0, 0}];steps, {1000}], _ ? (LessThan[100])]]
Wolfram Research (2021)，CarlsonRK，Wolfram 语言函数，https://reference.wolfram.com/language/ref/CarlsonRK.html.

文本

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_carlsonrk, author="Wolfram Research", title="{CarlsonRK}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/CarlsonRK.html}", note=[Accessed: 09-August-2026]}

BibLaTeX

@online{reference.wolfram_2026_carlsonrk, organization={Wolfram Research}, title={CarlsonRK}, year={2021}, url={https://reference.wolfram.com/language/ref/CarlsonRK.html}, note=[Accessed: 09-August-2026]}