EllipticLog
EllipticLog
EllipticLog[{x,y},{a,b}]
给出与椭圆曲线 
相关的广义对数.
更多信息
- 数学函数，适宜于符号和数值运算.
- EllipticLog[{x,y},{a,b}] 被定义为积分
的值，其中，平方根的符号通过给出使得 
成立的 y 值指定. - EllipticLog 可求任意数值精度的值.
范例
打开所有单元 关闭所有单元基本范例 (2)
范围 (16)
数值计算 (4)
EllipticLog[{2, 5}, {3, 2.5}]EllipticLog[{1, 2}, {1, 2.}]N[EllipticLog[{1, 3}, {2, 6}], 25]EllipticLog[{1.000000000000000000000, 3}, {2, 6}]EllipticLog[{2 + I, -I Sqrt[15]}, {1 - I, -12.0}]EllipticLog[{2, 5`60}, {3, 5 / 2}]//TimingEllipticLog[{2`100000, 5}, {3, 5 / 2}];//Timing特殊值 (3)
With[{x = 0.3}, EllipticLog[{x, Sqrt[x ^ 3 - 3x ^ 2 + x]}, {-3, 1}]]EllipticLog[{0, 0}, {0, 1}]//N求当 Abs[EllipticLog[{x,Sqrt[x^3+5x^2+x]},{5,1}]]=0.8 时，
的值：
xval = x /. FindRoot[Abs[EllipticLog[{x, Sqrt[x ^ 3 + 5x ^ 2 + x]}, {5, 1}]] == 0.8, {x, 0.5}]Plot[Abs[EllipticLog[{x, Sqrt[x ^ 3 + 5x ^ 2 + x]}, {5, 1}]], {x, -2, 4}, Epilog -> Style[Point[{xval, Abs[EllipticLog[{xval, Sqrt[xval ^ 3 + 5xval ^ 2 + xval]}, {5, 1}]]}], PointSize[Large], Red]]可视化 (2)
绘制 EllipticLog 的函数：
Plot[{EllipticLog[{u, Sqrt[u ^ 3 + (1 / 2)u]}, {0, 1 / 2}], EllipticLog[{u, Sqrt[u ^ 3 + 2 u ^ 2]}, {2, 0}], EllipticLog[{u, Sqrt[u ^ 3 + 5 u ^ 2 + u]}, {5, 1}]}, {u, 0, 3}]绘制 EllipticLog[{z,Sqrt[z^3+2 z^2+ z]},{2,1}] 实部：
ComplexContourPlot[Re[EllipticLog[{z, Sqrt[(z) ^ 3 + 2 (z) ^ 2 + z]}, {2, 1}]], {z, -4 - 4I, 4 + 4 I}, IconizedObject[«PlotOptions»]]绘制 EllipticLog[{x+ y,Sqrt[z^3+2 z^2+ z]},{2,1}] 虚部：
ComplexContourPlot[Im[EllipticLog[{z, Sqrt[(z) ^ 3 + 2 (z) ^ 2 + z]}, {2, 1}]], {z, -4 - 4I, 4 + 4 I}, IconizedObject[«PlotOptions»]]函数的属性 (3)
EllipticLog 不是解析函数：
FunctionAnalytic[EllipticLog[{x, y}, {a, b}], {a, b, x, y}]FunctionSingularities[EllipticLog[{x, y}, {a, b}], {a, b, x, y}]FunctionDiscontinuities[EllipticLog[{x, y}, {a, b}], {a, b, x, y}]//Quiet
FunctionSign[EllipticLog[{u, Sqrt[u ^ 3 + (1 / 2)u]}, {0, 1 / 2}], u]
FunctionConvexity[EllipticLog[{u, Sqrt[u ^ 3 + (1 / 2)u]}, {0, 1 / 2}], u]微分 (2)
D[EllipticLog[{x, y}, {a, b}], x]用 Integrate 计算不定积分：
Integrate[EllipticLog[{x, y}, {a, b}], x]FullSimplify[D[%, x]]级数展开 (2)
用 Series 求泰勒展开式：
Series[EllipticLog[{x, Sqrt[x ^ 3 + c x ^ 2 + d x]}, {c, d}], {x, 1, 2}]//Normal//FullSimplify
terms = Normal@Table[Series[EllipticLog[{x, Sqrt[x ^ 3 + x ^ 2 + 2 x]}, {1, 2}], {x, 1, m}], {m, 1, 5, 2}];
Plot[{EllipticLog[{x, Sqrt[x ^ 3 + x ^ 2 + 2 x]}, {1, 2}], terms}, {x, -5, 5}, PlotRange -> {-5, 5}]Series[EllipticLog[{x, Sqrt[x ^ 3 + c x ^ 2 + d x]}, {c, d}], {x, x0, 2}]//Normal// FullSimplify应用 (2)
ellipticMultiply[{{x1_, y1_}, {x2_, y2_}}, {a_, b_}] :=
Module[{s = (y2 - y1) / (x2 - x1), x3, y3},
x3 = s ^ 2 - x1 - x2 - a;
y3 = -y1 + s(x1 - x3);{x3, y3}]p1 = EllipticExp[-1 / 3., {4, 1}];
p2 = EllipticExp[1 / 2., {4, 1}];p3 = ellipticMultiply[{p1, p2}, {4, 1}]在乘积点的 EllipticLog 值等于对应因子的 EllipticLog 值之和：
EllipticLog[p3, {4, 1}] == EllipticLog[p1, {4, 1}] + EllipticLog[p2, {4, 1}]用 EllipticLog 来表达 CarlsonRF:
With[{x = 1, a = 2, b = 6},
N[{EllipticLog[{x, Sqrt[x^3 + a x^2 + b x]}, {a, b}], -CarlsonRF[x, x + (a + Sqrt[a^2 - 4b]/2), x + (2b/a + Sqrt[a^2 - 4b])]}, 25]]属性和关系 (3)
D[EllipticLog[{x, y}, {a, b}], x]EllipticExp 和 EllipticLog 是互为相反的函数：
EllipticExp[0.5, {3, 4}]EllipticLog[%, {3, 4}]EllipticLog 与 InverseWeierstrassP 函数有密切的关系：
ellipticLog[{x_, y_}, {a_, b_}] := (1/2)InverseWeierstrassP[{(1/4)(x + (a/3)), (y/4)}, {(1/4)((a^2/3) - b), (1/8)(a/3)((b/2) - ((a/3))^2)}]a = 3;b = 2;
{x, y} = EllipticExp[1.2, {a, b}];ellipticLog[{x, y}, {a, b}]EllipticLog[{x, y}, {a, b}]相关链接
MathWorld
Wolfram 函数网站
Wolfram Research (1988)，EllipticLog，Wolfram 语言函数，https://reference.wolfram.com/language/ref/EllipticLog.html.
文本
Wolfram Research (1988)，EllipticLog，Wolfram 语言函数，https://reference.wolfram.com/language/ref/EllipticLog.html.
CMS
Wolfram 语言. 1988. "EllipticLog." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/EllipticLog.html.
APA
Wolfram 语言. (1988). EllipticLog. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/EllipticLog.html 年