CarlsonRE
![TemplateBox[{x, y}, CarlsonRE]](Files/CarlsonRE.zh/4.png "TemplateBox[{x, y}, CarlsonRE]"){kind=small}
CarlsonRE
CarlsonRE[x,y]
给出 Carlson 椭圆积分 ![TemplateBox[{x, y}, CarlsonRE]](Files/CarlsonRE.zh/1.png "TemplateBox[{x, y}, CarlsonRE]"){kind=small}
.
![TemplateBox[{x, y}, CarlsonRE]=1/piint_0^infty(t+x)^(-1/2)(t+y)^(-1/2)sqrt(t)( x/(t+x)+y/(t+y))dt](Files/CarlsonRE.zh/2.png "TemplateBox[{x, y}, CarlsonRE]=1/piint_0^infty(t+x)^(-1/2)(t+y)^(-1/2)sqrt(t)( x/(t+x)+y/(t+y))dt"){kind=small}
范例
打开所有单元 关闭所有单元基本范例 (3)
CarlsonRE[3., 5.]Plot3D[CarlsonRE[x, y], {x, 0, 3}, {y, 0, 3}]CarlsonRE 与第二类勒让德完全椭圆积分相关：
(Pi/2)CarlsonRE[1, 1 - m] /. m -> 0.75EllipticE[0.75]范围 (12)
数值计算 (5)
CarlsonRE[1.0, 1 / GoldenRatio]N[CarlsonRE[2, 7], 50]CarlsonRE[2, 7.67890123456789012345]CarlsonRE[2, 7.678901234567890123456789012345]CarlsonRE[2. + 0.5 I, 2. - 0.5I]Timing[CarlsonRE[11, 3`500]]Timing[Precision[CarlsonRE[11, 3`100000]]]CarlsonRE 以元素方式线性作用于列表：
CarlsonRE[{1, 2, 3, 4, 5}, {5., 4., 3., 2., 1.}]导数和积分 (2)
函数表示 (1)
TraditionalForm 格式：
CarlsonRE[x, y]//TraditionalForm函数恒等和化简 (3)
CarlsonRE 满足欧拉-泊松偏微分方程：
(x - y)Subscript[∂, xy]CarlsonRE[x, y] + (1/2) Subscript[∂, y]CarlsonRE[x, y] - (1/2)Subscript[∂, x]CarlsonRE[x, y] == 0//FunctionExpand//FullSimplifyCarlsonRE 满足欧拉齐次关系：
x Subscript[∂, x]CarlsonRE[x, y] + y Subscript[∂, y]CarlsonRE[x, y] == (1/2) CarlsonRE[x, y]//FunctionExpand//FullSimplifyCarlsonRE 满足的偏微分方程：
Subscript[∂, x]CarlsonRE[x, y] + Subscript[∂, y]CarlsonRE[x, y] == (1/2) CarlsonRK[x, y]//FunctionExpand//FullSimplify应用 (3)
With[{a = Sqrt[2], b = 1},
N[2π CarlsonRE[a^2, b^2], 20]]与 ArcLength 的结果对比：
With[{a = Sqrt[2], b = 1},
ArcLength[{a Cos[t], b Sin[t]}, {t, 0, 2π}, WorkingPrecision -> 20]]二次形式平方根的期望值，与正态分布相关：
MatrixForm[mat = HilbertMatrix[2]]NExpectation[Sqrt[{u, v}.mat.{u, v} / 2], {u, v}BinormalDistribution[0]]//Quiet与关于 CarlsonRE 的闭合形式比较：
N[Gamma[(3/2)](CarlsonRE@@Eigenvalues[mat]), 25]R = 1 / 6;r = 5 / 9;b = 5 / 11;
cyl = Cylinder[{{0, 0, 0}, {0, 0, 2r}}, R];
ball = Ball[{b, 0, r}, r];Show@{
Graphics3D[{Opacity[1 / 2], cyl, ball}], Region[RegionIntersection[cyl, ball], PlotTheme -> "Web"]}用关于 Carlson 积分表示的圆柱与球相交区域的体积：
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 的结果相比较：:
Volume[RegionIntersection[cyl, ball], WorkingPrecision -> 20]属性和关系 (2)
CarlsonRE 在其参数排列置换后不变：
CarlsonRE[x, y] == CarlsonRE[y, x]CarlsonRE 和 CarlsonRK 满足勒让德关系：
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}}
Wolfram Research (2021)，CarlsonRE，Wolfram 语言函数，https://reference.wolfram.com/language/ref/CarlsonRE.html.
文本
Wolfram Research (2021)，CarlsonRE，Wolfram 语言函数，https://reference.wolfram.com/language/ref/CarlsonRE.html.
CMS
Wolfram 语言. 2021. "CarlsonRE." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/CarlsonRE.html.
APA
Wolfram 语言. (2021). CarlsonRE. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/CarlsonRE.html 年