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JacobiAmplitude[u,m]

给出雅可比 (Jacobi) 椭圆函数的幅值 TemplateBox[{u, m}, JacobiAmplitude].

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范例  
基本范例  
范围  
数值运算  
特殊值  
可视化  
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函数属性  
微分  
级数展开式  
函数表示  
应用  
属性和关系  
可能存在的问题  
参见
技术笔记
相关指南
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历史
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JacobiAmplitude

JacobiAmplitude[u,m]

给出雅可比 (Jacobi) 椭圆函数的幅值 TemplateBox[{u, m}, JacobiAmplitude].

更多信息

  • 数学函数，适合符号和数值运算.
  • JacobiAmplitude[u,m] 把一个椭圆函数的自变量 u 转换成幅值 ϕ.
  • JacobiAmplitude 是第一类型的椭圆积分的逆运算. 如果 u=TemplateBox[{phi, m}, EllipticF]，则 phi=TemplateBox[{u, m}, JacobiAmplitude].
  • 对于每对整数 和 ，JacobiAmplitude[u,m] 在复平面 u 上有一个分支切割断点，从 2 s TemplateBox[{m}, EllipticK]+ⅈ (4 t+1) TemplateBox[{{1, -, m}}, EllipticK] 到 2 s TemplateBox[{m}, EllipticK]+ⅈ (4 t+3) TemplateBox[{{1, -, m}}, EllipticK].
  • 对于某些特定变量值，JacobiAmplitude 自动运算出精确值.
  • JacobiAmplitude 可计算到任意数值精度.
  • JacobiAmplitude 自动逐项作用于列表.

范例

打开所有单元 关闭所有单元

基本范例  (4)

数值运算：

Wolfram Language code: JacobiAmplitude[4., 2 / 3]

在实数的子集上绘图：

Wolfram Language code: Plot[JacobiAmplitude[x, 2 / 3], {x, -4, 4}]

在复数的子集上绘图：

Wolfram Language code: ComplexPlot3D[JacobiAmplitude[z, 1 / 3], {z, -2 - 2I, 2 + 2I}, PlotLegends -> Automatic]

在原点的级数展开：

Wolfram Language code: Series[JacobiAmplitude[z, m], {z, 0, 6}]
Wolfram Language code: Series[JacobiAmplitude[z, m], {m, 0, 1}]

范围  (26)

数值运算  (5)

高精度运算：

Wolfram Language code: N[JacobiAmplitude[Pi / 3, 1 / 5], 50]

输出精度与输入精度一致：

Wolfram Language code: JacobiAmplitude[Pi / 3, 0.200000000000000000000000000]

对复变量求值：

Wolfram Language code: JacobiAmplitude[0.2 + 0.1 I, 0.2 I]

用高精度高效评估 JacobiAmplitude：

Wolfram Language code: JacobiAmplitude[Pi / 3, 0.2`500]//Timing
Wolfram Language code: JacobiAmplitude[Pi / 3, 0.2`10000];//Timing

使用 Around 计算平均值统计区间：

Wolfram Language code: JacobiAmplitude[ Around[2 / 3, 0.001], 1 / 3]

计算数组的元素值：

Wolfram Language code: JacobiAmplitude[{{1 / 2, -1}, {-5 / 3, 1 / 2}}, 1]

或使用 MatrixFunction 计算矩阵 JacobiAmplitude 函数：

Wolfram Language code: MatrixFunction[JacobiAmplitude[#, 1]&, {{1 / 2, -1}, {-5 / 3, 1 / 2}}]//FullSimplify

特殊值  (4)

自动产生简化的精确答案：

Wolfram Language code: {JacobiAmplitude[x, 0], JacobiAmplitude[x, 1]}
Wolfram Language code: {JacobiAmplitude[0, m], JacobiAmplitude[EllipticK[m], m]}
Wolfram Language code: JacobiAmplitude[EllipticK[m] / 2 + I EllipticK[1 - m] / 2, m]

某些特殊情况无穷处的值：

Wolfram Language code: {JacobiAmplitude[Infinity, 0], JacobiAmplitude[Infinity, 1]}

求 TemplateBox[{x, {-, 3}}, JacobiAmplitude]⩵1 的根：

Wolfram Language code: xzero = x /. FindRoot[JacobiAmplitude[x, -3] == 1, {x, 2}]
Wolfram Language code: Plot[JacobiAmplitude[x, -3], {x, -3, 6}, WorkingPrecision -> 10, Epilog -> Style[Point[{xzero, JacobiAmplitude[xzero, -3]}], PointSize[Large], Red]]

自动应用奇偶校验：

Wolfram Language code: JacobiAmplitude[-u, m]

可视化  (3)

绘制各种参数值 的 JacobiAmplitude 函数：

Wolfram Language code: Plot[{JacobiAmplitude[x, -1], JacobiAmplitude[x, 1 / 2], JacobiAmplitude[x, 1], JacobiAmplitude[x, 2]}, {x, -4, 4}, PlotLegends -> "Expressions"]

按照参数 的函数绘制 JacobiAmplitude：

Wolfram Language code: Plot[{JacobiAmplitude[1, m], JacobiAmplitude[2, m], JacobiAmplitude[3, m]}, {m, -10, 10}, PlotLegends -> "Expressions"]

绘制 TemplateBox[{z, 2}, JacobiAmplitude] 的实部：

Wolfram Language code: ComplexContourPlot[Re[JacobiAmplitude[z, 2]], {z, -6 - 6I, 6 + 6I}, Contours -> 20]

绘制 TemplateBox[{z, 2}, JacobiAmplitude] 的虚部：

Wolfram Language code: ComplexContourPlot[Im[JacobiAmplitude[z, 2]], {z, -6 - 6I, 6 + 6I}, IconizedObject[«PlotOptions»]]

函数属性  (5)

TemplateBox[{x, {2, /, 3}}, JacobiAmplitude] 是 x 的解析函数：

Wolfram Language code: FunctionAnalytic[JacobiAmplitude[x, 2 / 3], x]

该函数没有奇点也没有断点：

Wolfram Language code: FunctionSingularities[JacobiAmplitude[x, 2 / 3], x]
Wolfram Language code: FunctionDiscontinuities[JacobiAmplitude[x, 2 / 3], x]

TemplateBox[{x, {2, /, 3}}, JacobiAmplitude] 为单射函数：

Wolfram Language code: FunctionInjective[JacobiAmplitude[x, 2 / 3], x]
Wolfram Language code: Plot[{JacobiAmplitude[x, 2 / 3], 3}, {x, -5, 5}]

TemplateBox[{x, {2, /, 3}}, JacobiAmplitude] 是满射函数：

Wolfram Language code: FunctionSurjective[JacobiAmplitude[x, 2 / 3], x]
Wolfram Language code: Plot[{JacobiAmplitude[x, 2 / 3], -5}, {x, -10, 10}]

JacobiAmplitude 不是非负也不是非正函数：

Wolfram Language code: FunctionSign[JacobiAmplitude[x, 2 / 3], x]

JacobiAmplitude 不是凸函数也不是凹函数：

Wolfram Language code: FunctionConvexity[JacobiAmplitude[x, 1 / 3], x]

微分  (3)

一阶导数：

Wolfram Language code: D[JacobiAmplitude[x, m], x]

高阶导数：

Wolfram Language code: derivs = Table[D[JacobiAmplitude[x, m], {x, n}], {n, 1, 4}]//FullSimplify

绘制 时的高阶导数：

Wolfram Language code: Plot[Evaluate[derivs /. m -> 2 / 3], {x, -5, 5}, PlotLegends -> {"First Derivative", "Second Derivative", "Third Derivative", "Fourth Derivative"}]

关于 的导数：

Wolfram Language code: D[JacobiAmplitude[x, m], m]

级数展开式  (3)

TemplateBox[{x, {2, /, 3}}, JacobiAmplitude] 的泰勒展开式：

Wolfram Language code: Series[JacobiAmplitude[x, 2 / 3], {x, 0, 7}]

绘制 附近 TemplateBox[{x, {2, /, 3}}, JacobiAmplitude] 的前 3 个近似式：

Wolfram Language code: terms = Normal@Table[Series[JacobiAmplitude[x, 2 / 3], {x, 0, n}], {n, 1, 5, 2}]; Plot[{JacobiAmplitude[x, 2 / 3], terms}, {x, -2, 2}]

TemplateBox[{1, m}, JacobiAmplitude] 的泰勒展开式：

Wolfram Language code: Series[JacobiAmplitude[1, m], {m, 0, 3}]

绘制 附近 TemplateBox[{1, m}, JacobiAmplitude] 的前 3 个近似式：

Wolfram Language code: terms = Normal@Table[Series[JacobiAmplitude[1, m], {m, 0, n}], {n, 1, 3}]; Plot[{JacobiAmplitude[1, m], terms}, {m, -9, 9}]

JacobiAmplitude 可应用于幂级数：

Wolfram Language code: JacobiAmplitude[u, Log[1 + m] + O[m] ^ 3]
Wolfram Language code: JacobiAmplitude[u Cos[u] + O[u] ^ 10, m]//Simplify

函数表示  (3)

JacobiAmplitude 是 EllipticF 的反函数：

Wolfram Language code: Solve[u == EllipticF[ϕ, m], ϕ]//Quiet

积分表示：

Wolfram Language code: Integrate[JacobiDN[t, m], {t, 0, u}]

TraditionalForm 格式：

Wolfram Language code: JacobiAmplitude[u, m]//TraditionalForm

应用  (4)

过冲模式下的单摆方程的解：

Wolfram Language code: ϕ[t_] = With[{k = 1 + 10^-3}, 2JacobiAmplitude[k t, k^-2]];

代入单摆方程进行验证：

Wolfram Language code: ϕ''[t] + Sin[ϕ[t]]//FullSimplify

绘制解：

Wolfram Language code: Plot[ϕ[t], {t, 0, 36}]

sine‐Gordon 方程的相对论解：

Wolfram Language code: u[t_, x_, {v_, k_}] = Piecewise[{{π - 2JacobiAmplitude[(x - v t/kSqrt[1 - v^2]), k^2], k ≤ 1}, {2 (π - JacobiAmplitude[((1 + k) (x - v t)/2 k Sqrt[1 - v^2]), (4 k/(1 + k)^2)] + JacobiAmplitude[((1 + k) (x - v t)/2 k Sqrt[1 - v^2]) - (1 + (1/k)) EllipticK[(1/k^2)], (4 k/(1 + k)^2)]), k > 1}}];

代入 sine‐Gordon 方程，验证解：

Wolfram Language code: Assuming[k < 1, Subscript[∂, {t, 2}]u[t, x, {v, k}] - Subscript[∂, {x, 2}]u[t, x, {v, k}] + Sin[u[t, x, {v, k}]] //FullSimplify]

绘制 和 取不同值时的解：

Wolfram Language code: Plot3D[u[t, x, {0.95, 0.99}], {x, -4, 4}, {t, 0, 2}]
Wolfram Language code: Plot3D[u[t, x, {0.8, 1.2}], {x, -4, 4}, {t, 0, 2}]

形成和绘制广义的傅立叶级数：

Wolfram Language code: Function[m, Plot[Evaluate[Table[Sum[-(-1) ^ k Sin[k JacobiAmplitude[2 EllipticK[m] / Pi v, m]] / k, {k, o}], {o, 8}]], {v, -2Pi, 2Pi}, ImageSize -> Small]] /@ {0, 0.33, 0.66, 0.99}

椭圆参数 以及 () 形成的球面三角形：

Wolfram Language code: SphericalTriangleAnglesAndSides[m_, {u1_, u2_, u3_}] /; inputQ[m, {u1, u2, u3}] := {JacobiAmplitude[2EllipticK[m]{u1, u2, u3}, m], ArcSin[Sqrt[m]JacobiSN[2EllipticK[m]{u1, u2, u3}, m]]}
Wolfram Language code: inputQ[m_, us_List] := 0 < m < 1 && Min[us] > 0 && Total[us] == 1

顶点 、 和 处的角及与每个顶点相对的边：

Wolfram Language code: {{∠A, ∠B, ∠C}, {a, b, c}} = SphericalTriangleAnglesAndSides[0.7, {0.2, 0.3, 0.5}]

验证 Cagnoli 方程，它与球面三角形的所有角度和边的测量有关：

Wolfram Language code: Sin[b]Sin[c] + Cos[b]Cos[c]Cos[∠A] == Sin[∠B]Sin[∠C] - Cos[∠B]Cos[∠C]Cos[a]

计算三角形的面积，也称为球面角盈：

Wolfram Language code: Total[{∠A, ∠B, ∠C}] - Pi

用 L'Huilier 定理 [MathWorld] 计算球面角盈：

Wolfram Language code: With[{𝓈 = Total[{a, b, c}] / 2}, 4ArcTan[Sqrt[Tan[(𝓈/2)]Tan[(𝓈 - a/2)]Tan[(𝓈 - b/2)]Tan[(𝓈 - c/2)]]]]

计算 3D 单位球面上的点：

Wolfram Language code: pC = {1, 0, 0}; normalBC = {0, 0, 1}; pB = RotationTransform[a, normalBC][pC]; normalAB = RotationTransform[-∠B, pB][normalBC]; pA = RotationTransform[-c, normalAB][pB];

检查求出的点和边之间的一致性：

Wolfram Language code: {{(pC.pB/Norm[pC] Norm[pB]), Cos[a]}, {(pA.pC/Norm[pA] Norm[pC]), Cos[b]}, {(pA.pB/Norm[pA] Norm[pB]), Cos[c]}}

可视化该三角形：

Wolfram Language code: Show[ Graphics3D[{{Opacity[1 / 10, Gray], Sphere[{0, 0, 0}, 1]}, {Black, Sphere[{0, 0, 0}, 1 / 50], Arrowheads[Medium], Arrow[{{0, 0, 0}, pA}], Arrow[{{0, 0, 0}, pB}], Arrow[{{0, 0, 0}, pC}] }}], {ParametricPlot3D[RotationTransform[a t, {pB, pC}][pB]//Evaluate, {t, 0, 1}], ParametricPlot3D[RotationTransform[b t, {pA, pC}][pA]//Evaluate, {t, 0, 1}], ParametricPlot3D[RotationTransform[c t, {pA, pB}][pA]//Evaluate, {t, 0, 1}]}]

属性和关系  (5)

与反函数组合：

Wolfram Language code: {EllipticF[JacobiAmplitude[z, m], m], JacobiAmplitude[EllipticF[z, m], m]}

用 PowerExpand 略去反函数的多值性：

Wolfram Language code: PowerExpand[%]

将三角函数应用到 JacobiAmplitude：

Wolfram Language code: {Cos[JacobiAmplitude[z, m]], Sin[JacobiAmplitude[z, m]]}

求解一个超越方程：

Wolfram Language code: Solve[JacobiAmplitude[x, m]^2 + 3JacobiAmplitude[x, m] == a, x]//Quiet

从一个微分方程中获得：

Wolfram Language code: DSolve[Subscript[∂, z]w[z] - JacobiDN[z, m] == 0, w[z], z]

对于任意整数 和 ，JacobiAmplitude 有一个分支切割，从 2 s TemplateBox[{m}, EllipticK]+ⅈ (4 t+1) TemplateBox[{{1, -, m}}, EllipticK] 到 2 s TemplateBox[{m}, EllipticK]+ⅈ (4 t+3) TemplateBox[{{1, -, m}}, EllipticK]：

Wolfram Language code: Table[JacobiAmplitude[2s EllipticK[m] + I (4t + 1)EllipticK[1 - m], m], {s, -2, 2}, {t, -2, 2}]//Flatten//Union
Wolfram Language code: Table[JacobiAmplitude[2s EllipticK[m] + I (4t + 3)EllipticK[1 - m], m], {s, -2, 2}, {t, -2, 2}]//Flatten//Union

可视化不同模数的分支切割：

Wolfram Language code: ShowJacobiAmplitudeBranchCuts[m_ ? NumericQ] := Module[{K, Kp, rK, iK, rKp, iKp}, K = EllipticK[m];Kp = EllipticK[1 - m]; {rK, iK} = ReIm[K];{rKp, iKp} = ReIm[Kp]; Graphics[{Brown, Line[Flatten[Table[{{2s rK - (4t + 1)iKp, 2s iK + (4t + 1)rKp}, {2s rK - (4t + 3)iKp, 2s iK + (4t + 3)rKp}}, {s, -3, 3}, {t, -2, 2}], 1]]}, Frame -> True, AspectRatio -> 1, ImageSize -> Small, GridLines -> Automatic, GridLinesStyle -> LightGray] ]
Wolfram Language code: {ShowJacobiAmplitudeBranchCuts[-3], ShowJacobiAmplitudeBranchCuts[1 / 3], ShowJacobiAmplitudeBranchCuts[3], ShowJacobiAmplitudeBranchCuts[3 + I]}

可能存在的问题  (1)

对于整数 ， 时 TemplateBox[{u, m}, JacobiAmplitude] 的分支切割与实轴相交于 2(2 s+1) TemplateBox[{{1, /, m}}, EllipticK]/sqrt(m) 处：

Wolfram Language code: With[{m = 3}, branchPoints = {PointSize[Medium], Point[Table[{2(2s + 1)EllipticK[1 / m] / Sqrt[m], 0}, {s, -2, 2}]]}; Plot[JacobiAmplitude[u, m], {u, -6, 6}, Epilog -> branchPoints]]

对于 ，需要幅值连续的物理应用，请使用以下定义：

Wolfram Language code: am[u_, m_] := -(π/2) + JacobiAmplitude[(1/2) (1 + Sqrt[m]) u, (4 Sqrt[m]/(1 + Sqrt[m])^2)] - JacobiAmplitude[(1/2) (1 + Sqrt[m]) u - (1 + (1/Sqrt[m])) EllipticK[(1/m)], (4 Sqrt[m]/(1 + Sqrt[m])^2)];

对于实数 且 ，上述函数与 sin^(-1)(TemplateBox[{u, m}, JacobiSN]) 一致：

Wolfram Language code: With[{m = 3}, Plot[{ArcSin[JacobiSN[u, m]], am[u, m]}, {u, -6, 6}, IconizedObject[«Plot options»]]]
Wolfram Research (1988)，JacobiAmplitude，Wolfram 语言函数，https://reference.wolfram.com/language/ref/JacobiAmplitude.html （更新于 2020 年）.

文本

Wolfram Research (1988)，JacobiAmplitude，Wolfram 语言函数，https://reference.wolfram.com/language/ref/JacobiAmplitude.html （更新于 2020 年）.

CMS

Wolfram 语言. 1988. "JacobiAmplitude." Wolfram 语言与系统参考资料中心. Wolfram Research. 最新版本 2020. https://reference.wolfram.com/language/ref/JacobiAmplitude.html.

APA

Wolfram 语言. (1988). JacobiAmplitude. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/JacobiAmplitude.html 年

BibTeX

@misc{reference.wolfram_2026_jacobiamplitude, author="Wolfram Research", title="{JacobiAmplitude}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiAmplitude.html}", note=[Accessed: 31-October-2026]}

BibLaTeX

@online{reference.wolfram_2026_jacobiamplitude, organization={Wolfram Research}, title={JacobiAmplitude}, year={2020}, url={https://reference.wolfram.com/language/ref/JacobiAmplitude.html}, note=[Accessed: 31-October-2026]}