EllipticNomeQ
EllipticNomeQ
给出对应于椭圆函数中参数 m 的 nome q.
更多信息
- 数学函数，适宜于符号和数值运算.
- EllipticNomeQ 与 EllipticK 相关，
![TemplateBox[{m}, EllipticNomeQ]=exp[-piTemplateBox[{{1, -, m}}, EllipticK]/TemplateBox[{m}, EllipticK]]](Files/EllipticNomeQ.zh/1.png "TemplateBox[{m}, EllipticNomeQ]=exp[-piTemplateBox[{{1, -, m}}, EllipticK]/TemplateBox[{m}, EllipticK]]")
. - EllipticNomeQ[m] 在复平面 m 上有一条从
到 
的分支线. - 对于某些特殊参数，EllipticNomeQ 自动算出精确值.
- EllipticNomeQ 可求任意精度的值.
- EllipticNomeQ 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 (5)
EllipticNomeQ[-2.]Plot[EllipticNomeQ[x], {x, -2, 2}]ComplexPlot3D[EllipticNomeQ[z], {z, -2 - 2I, 2 + 2I}, PlotLegends -> Automatic]Series[EllipticNomeQ[m], {m, 0, 5}]Infinity 处的级数展开式：
Series[EllipticNomeQ[x], {x, ∞, 2}]//Normal//FullSimplify范围 (30)
数值计算 (6)
EllipticNomeQ[.5]N[EllipticNomeQ[5 / 16]]N[EllipticNomeQ[3 / 17], 50]EllipticNomeQ[0.3333333333333333333]N[EllipticNomeQ[5 / 16 + I]]EllipticNomeQ[1 / 7`100]//TimingEllipticNomeQ[5 / 9`10000];//Timing用 Interval 和 CenteredInterval 对象计算最坏情况下的区间：
EllipticNomeQ[Interval[{0.5, .6}]]EllipticNomeQ[CenteredInterval[.5, 0.2]]或用 Around 计算一般情况下的统计区间：
EllipticNomeQ[ Around[.2, 0.01]]EllipticNomeQ[{{1 / 2, 1}, {0, 1 / 2}}]或用 MatrixFunction 计算矩阵形式的 EllipticNomeQ 函数：
MatrixFunction[EllipticNomeQ, {{1 / 2, 1}, {0, 1 / 2}}]//FullSimplify特殊值 (5)
Table[EllipticNomeQ[x], {x, -1, 1}]EllipticNomeQ[x]//FunctionExpandEllipticNomeQ[0]EllipticNomeQ[1 / 2]求当 EllipticNomeQ[x]=0.1 时 x 的值：
xval = x /. FindRoot[EllipticNomeQ[x] == 0.1, {x, 0.5}]Plot[EllipticNomeQ[x], {x, -2, 2}, Epilog -> Style[Point[{xval, EllipticNomeQ[xval]}], PointSize[Large], Red]]可视化 (2)
绘制各种参数值的 EllipticNomeQ 函数：
Plot[EllipticNomeQ[u], {u, -2, 2}]![TemplateBox[{z}, EllipticNomeQ]](Files/EllipticNomeQ.zh/2.png "TemplateBox[{z}, EllipticNomeQ]"){kind=small}
ComplexContourPlot[Re[EllipticNomeQ[z]], {z, -3 - 3I, 3 + 3 I}, Contours -> 24]![TemplateBox[{z}, EllipticNomeQ]](Files/EllipticNomeQ.zh/3.png "TemplateBox[{z}, EllipticNomeQ]"){kind=small}
ComplexContourPlot[Im[EllipticNomeQ[z]], {z, -3 - 3I, 3 + 3 I}, Contours -> 24]函数的属性 (10)
EllipticNomeQ 的实数和复数定义域：
FunctionDomain[EllipticNomeQ[x], x]FunctionDomain[EllipticNomeQ[z], z, Complexes]FunctionRange[EllipticNomeQ[x], x, y]//QuietEllipticNomeQ 逐项作用于列表的各个元素：
EllipticNomeQ[{0.2, 0.3, 0.4}]EllipticNomeQ 不是解析函数：
FunctionAnalytic[EllipticNomeQ[x], x]FunctionSingularities[EllipticNomeQ[x], x]//QuietFunctionDiscontinuities[EllipticNomeQ[x], x]//QuietEllipticNomeQ 在实定义域上非递减：
FunctionMonotonicity[{EllipticNomeQ[x], x < 1}, x]EllipticNomeQ 是单射函数：
FunctionInjective[EllipticNomeQ[x], x]Plot[{EllipticNomeQ[x], .1}, {x, -2, 2}]EllipticNomeQ 不是满射函数：
FunctionSurjective[EllipticNomeQ[x], x]//QuietPlot[{EllipticNomeQ[x], -.5}, {x, -2, 2}]EllipticNomeQ 既不是非负，也不是非正：
FunctionSign[{EllipticNomeQ[x], x < 1}, x]EllipticNomeQ 在实定义域上是凸函数：
FunctionConvexity[{EllipticNomeQ[x], x < 1}, x]TraditionalForm 格式：
EllipticNomeQ[m] // TraditionalForm微分 (2)
级数展开 (5)
用 Series 求泰勒展开式：
Series[EllipticNomeQ[x], {x, 0, 5}]
terms = Normal@Table[Series[EllipticNomeQ[x], {x, 0, m}], {m, 1, 5, 2}];
Plot[{EllipticNomeQ[x], terms}, {x, -5, 5}, PlotRange -> {-.5, .5}]求在 Infinity 处的级数展开式：
Series[EllipticNomeQ[x], {x, Infinity, 1}]//Normal//FullSimplify
Series[EllipticNomeQ[x], {x, DirectedInfinity[z], 1}, Assumptions -> x > 0]// Normal//FullSimplifySeries[EllipticNomeQ[x], {x, x0, 2}]//Normal// FullSimplifySeries[EllipticNomeQ[x], {x, 1, 2}] // FullSimplify // Normal // Quiet推广和延伸 (1)
将 EllipticNomeQ 应用于幂级数：
EllipticNomeQ[m - (m^2/2) + (m^3/9) + O[m]^4]应用 (3)
定义 Halphen 常数 [MathWorld]：
SetAttributes[HalphenConstant, Constant];
HalphenConstant/:N[HalphenConstant, wp_] := N[EllipticNomeQ[Root[{EllipticK[#] - 2EllipticE[#]&, 0.82611476598497033617737323860075640603409051645331}]], wp]x = N[HalphenConstant, 500]
Sum[(2k + 1)^2(-x)^k(k + 1) / 2, {k, 0, 50}]在复平面上绘制 EllipticNomeQ：
Plot3D[Im[EllipticNomeQ[x + I y]], {x, -2, 2}, {y, -2, 2}]ArithmeticGeometricMeanIteration[n_, a_, b_] := With[{z = EllipticNomeQ[1 - (b / a)^2]}, ArithmeticGeometricMean[a, b]{EllipticTheta[3, 0, z^2^n]^2, EllipticTheta[4, 0, z^2^n]^2}]Table[ArithmeticGeometricMeanIteration[n, 1., 2.], {n, 0, 4}]With[{a = N[1, 2000], b = N[2, 2000]}, Table[ArithmeticGeometricMeanIteration[n, a, b], {n, 0, 10}] - ArithmeticGeometricMean[a, b]]//SetPrecision[#, 3]&
With[{a = 1``1010, b = 1 / Sqrt[2], o = 9}, 2ArithmeticGeometricMeanIteration[o + 1, a, b][[1]] ^ 2 / (1 - Sum[2 ^ n ({1, -1}.(ArithmeticGeometricMeanIteration[n, a, b] ^ 2)), {n, 0, o}])]% - Pi属性和关系 (6)
用 FullSimplify 化简包含 EllipticNomeQ 的表达式：
EllipticNomeQ[m] == Exp[-π EllipticK[1 - m] / EllipticK[m]]//FullSimplify{InverseEllipticNomeQ[EllipticNomeQ[z]], EllipticNomeQ[InverseEllipticNomeQ[z]]}PowerExpand[%]D[EllipticNomeQ[q], q]Solve[EllipticNomeQ[z] + EllipticNomeQ[z]^2 == g, z]FindRoot[EllipticNomeQ[z] + EllipticNomeQ[z ^ 2] + z == -1, {z, 2 + I}]Neville theta 函数的特殊值含有 EllipticNomeQ：
NevilleThetaC[I EllipticK[1 - m], m]NevilleThetaS[EllipticK[m] + I EllipticK[1 - m], m]可能存在的问题 (1)
对大多数已命名的特殊函数，直接函数是单值函数，其反函数是多值函数. EllipticNomeQ 是一个多值函数，其反函数 InverseEllipticNomeQ 是单值函数. 因此下面的例子无论在何处都是正确的.
InverseEllipticNomeQ[EllipticNomeQ[z]]巧妙范例 (1)
EllipticNomeQ 的黎曼（Riemann）面：
ParametricPlot3D[{Re[InverseEllipticNomeQ[r Exp[I ϕ]]], Im[InverseEllipticNomeQ[r Exp[I ϕ]]], r Sin[ϕ]}, {r, 0, 0.4}, {ϕ, 0, 2Pi}, PlotStyle -> Opacity[0.66], BoxRatios -> {1, 1, 1}]相关链接
MathWorld
Wolfram 函数网站
Wolfram Research (1996)，EllipticNomeQ，Wolfram 语言函数，https://reference.wolfram.com/language/ref/EllipticNomeQ.html.
文本
Wolfram Research (1996)，EllipticNomeQ，Wolfram 语言函数，https://reference.wolfram.com/language/ref/EllipticNomeQ.html.
CMS
Wolfram 语言. 1996. "EllipticNomeQ." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/EllipticNomeQ.html.
APA
Wolfram 语言. (1996). EllipticNomeQ. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/EllipticNomeQ.html 年