WOLFRAM

Wolfram 语言与系统参考资料中心

CarlsonRC[x,y]

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

更多信息
Details and Options Details and Options
范例  
基本范例  
范围  
数值计算  
指定值  
导数和积分  
函数表示  
函数恒等和化简  
应用  
属性和关系  
可能存在的问题  
参见
相关指南
历史
按以下格式引用：

CarlsonRC

CarlsonRC[x,y]

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

更多信息

范例

打开所有单元 关闭所有单元

基本范例  (3)

数值计算：

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

绘制函数：

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

CarlsonRC 与 TemplateBox[{phi, m}, EllipticF] 其中 的特殊情况 相关：

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

范围  (13)

数值计算  (6)

用数字进行计算：

Wolfram Language code: CarlsonRC[5, 4.]

更高精度的计算：

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

输出的精度可以追溯输入的精度：

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

复数参数的计算：

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

高效的高精度计算：

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

CarlsonRC 以元素方式线性作用于列表：

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

CarlsonRC 可与 Interval 和 CenteredInterval 对象一起使用：

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]]

指定值  (2)

自动生成简单精确值：

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

使用 FunctionExpand 将 CarlsonRC 转换为基本函数：

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

导数和积分  (2)

关于 的 TemplateBox[{x, y}, CarlsonRC] 的导数：

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

关于 的 TemplateBox[{x, y}, CarlsonRC] 的导数：

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

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

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

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

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

函数表示  (1)

TraditionalForm 格式：

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

函数恒等和化简  (2)

CarlsonRC 满足欧拉-泊松偏微分方程：

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 满足欧拉齐次关系：

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

应用  (3)

使用 CarlsonRC 提供 CarlsonRF[x,y,z] 的上边界和下边界：

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 用于简洁表达 EllipticPi 的参数关系变化：

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.}}

使用 CarlsonRC 可表示 CarlsonRJ 的参数关系变化：

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.}}

属性和关系  (3)

对于 ，可用 ArcCos 表示 TemplateBox[{x, y}, CarlsonRC]：

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

如果 位于负实轴上，则 TemplateBox[{x, y}, CarlsonRC] 被解释为柯西主值：

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

与正参数的等价表达式进行比较：

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

通过 FunctionExpand 可用更简单的函数来表达 CarlsonRC：

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

可能存在的问题  (1)

一般来说，TemplateBox[{x, z, z}, CarlsonRF]=TemplateBox[{x, z}, CarlsonRC]；然而，由于这两个函数的解析结构不同，运算被限制在不位于负实数轴的 的数值上：

Wolfram Language code: CarlsonRF[x, z, z]
Wolfram Language code: % /. z -> Pi
Wolfram Research (2021)，CarlsonRC，Wolfram 语言函数，https://reference.wolfram.com/language/ref/CarlsonRC.html （更新于 2023 年）.

文本

Wolfram Research (2021)，CarlsonRC，Wolfram 语言函数，https://reference.wolfram.com/language/ref/CarlsonRC.html （更新于 2023 年）.

CMS

Wolfram 语言. 2021. "CarlsonRC." Wolfram 语言与系统参考资料中心. Wolfram Research. 最新版本 2023. https://reference.wolfram.com/language/ref/CarlsonRC.html.

APA

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

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-August-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-August-2026]}