NContourIntegrate
NContourIntegrate
NContourIntegrate[f,z∈cont]
给出 f 在复平面中由 cont 定义的环线上的数值积分.
更多信息和选项
- 围道积分也称为路径积分或复线积分.
- 围道积分出现在复分析中对全纯和亚纯函数的研究中，但现在它们的应用范围很广，包括计算拉普拉斯逆变换和 Z 变换、定积分与和，以及偏微分方程的解.
- 函数
沿环线 cont 
的围道积分由下式给出： - 围道积分的值与参数化无关，但与环线 cont 的方向有关.
- 函数 f 通常是 z 的亚纯函数，但它可以是在复平面中 cont 邻域中定义的任意分段连续函数.
- 可使用柯西留数定理计算亚纯函数
沿闭合环线 cont 的围道积分. - 常用的闭合环线 cont 包括： »
-
{"Hairpin",hl} 包围半直线 hl 
{"UpperSemicircle",ipts,epts} 包围上半平面，包括点 ipts，不包括点 epts，全部在实轴上 
{"LowerSemicircle",ipts,epts} 包围下半平面，包括点 ipts，不包括点 epts，全部在实轴上 
{"Dumbbell",pt1,pt2} 包围由点 pt1 和 pt2 给定的胶囊 - 复数点以 {x,y} 数据对的形式给出；复半直线以 HalfLine 基元的形式给出.
- 也可以用
中的曲线区域 (RegionQ) 指定 
中的环线 cont. - 对于参数化环线 ParametricRegion[{x[t],y[t]},{{t,a,b}}]，方向是 t 增大的方向.
中的特殊环线及认定的方向：-
Line[{p1,p2,…}] 从 p1 到 p2 等 
HalfLine[{p1,p1}] 从 p1 到 p2 
InfiniteLine[{p1,p2}] 从 p1 到 p2 
Circle[p,…] 逆时针 - 可以使用诸如 Polygon 之类的面区域，然后将环线视为边界环线 RegionBoundary[Polygon[…]].
中的特殊面区域及认定的边界环线的方向：-
Triangle[{p1,p2,p3}] 逆时针 
Rectangle[p1,p2] 逆时针 
RegularPolygon[n,…] 逆时针 
Polygon[{p1,p2,…}{{q1,q2,…},…}] 外部环线采用逆时针方向，内部环线采用顺时针方向 
Disk[p,…] 逆时针 
Ellipsoid[p,…] 逆时针 
StadiumShape[{p1,p2},r] 逆时针 
Annulus[p,{rm,rm},…] 外部环线采用逆时针方向，内部环线采用顺时针方向 - cont 中的区域可能被 Inactive 封装，以避免自动计算.
- 可给出以下选项：
-
AccuracyGoal Automatic 寻求的绝对准确度 MaxPoints Automatic 样本点的最大数量 MaxRecursion Automatic 递归子划分的最大数量 Method Automatic 要使用的方法 MinRecursion 0 递归子划分的最小数量 PrecisionGoal Automatic 寻求的精度 WorkingPrecision Automatic 内部计算使用的精度
范例
打开所有单元 关闭所有单元基本范例 (3)
NContourIntegrate[1 / z, z∈Circle[]]Show[ComplexPlot[(z^3 (1 - 5 z)/(1 + z) (1 + 2 z^4)), {z, -4 - 4I, 4 + 4I}], ...]NContourIntegrate[(z ^ 3(1 - 5z)/(1 + z) (1 + 2 z^4)), z∈Circle[{0, 0}, 3]]Show[ComplexPlot[(Cos[z]/z(z^2 + 3)(z - 2)), {z, -4 - 4I, 4 + 4I}], ...]NContourIntegrate[(Cos[z]/z(z^2 + 3)(z - 2)), z∈Circle[{0, 0}, {1, 3}]]与 ContourIntegrate 的结果相比较：
ContourIntegrate[(Cos[z]/z(z^2 + 3)(z - 2)), z∈Circle[{0, 0}, {1, 3}]]N[%]范围 (46)
基本用法 (9)
f = (1/(z - 1)^2 (z - I));ComplexPlot[f, {z, -3 - 3I, 3 + 3I}, ...]NContourIntegrate[f, z∈Circle[{1, 0}, 1]]与 ContourIntegrate 相比较：
ContourIntegrate[f, z∈Circle[{1, 0}, 1]]f = Exp[2z] / (z - 1);reg = Line[{{0, 0}, {1, 1}, {2, 0}}];ComplexPlot[f, {z, -1 - I, 3 + 3I}, ...]NContourIntegrate[f, z∈reg]f = Sin[Pi * z] ^ 2 / (z - 1 - I);reg = Disk[{0, 0}, 3, {0, Pi}];NContourIntegrate[f, z∈reg]f = Sin[z ^ 2 + 1 / z] ^ 3;reg = Circle[];NContourIntegrate[f, z∈reg]reg = ParametricRegion[{Cos[t], Sin[t]}, {{t, 0, 2Pi}}];NContourIntegrate[1 / z, z∈reg]f = (Cosh[z^2 - 1]/z^3 + I / 2);reg = Disk[{0, 0}, 1, {0, Pi}];ComplexPlot[f, {z, -2 - 2I, 2 + 2I}, ...]NContourIntegrate[f, z∈reg]f = (Cos[1 / z]/z);reg = Circle[];ComplexPlot[f, {z, -2 - 2I, 3 + 2I}, ...]NContourIntegrate[f, z∈reg]f = Exp[1 / Conjugate[z]];reg = Line[{{-1, -1}, {0, -1}, {1, -3 / 2}}];ComplexPlot[f, {z, -2 - 2I, 2 + 2I}, Epilog -> {RGBColor[0.33333333333333337, 0.33333333333333337, 0.33333333333333337], Arrow@reg}, ImageSize -> Small]NContourIntegrate[f, z∈reg]f = Log[z ^ 2 - 1 / 2];reg = Circle[];ComplexPlot[f, {z, -1.5 - 1.5I, 1.5 + 1.5I}, ...]NContourIntegrate[f, z∈reg, AccuracyGoal -> 5]特殊主题：有理函数 (8)
NContourIntegrate[(3z ^ 3 + 2) / ((z - 1)(z ^ 2 + 9)), z∈Circle[{0, 0}, 4]]ComplexPlot[((z ^ 5 + 1) ^ 8/(z - I / 2)(z - 1 / 3) ^ 2), {z, -1 - I, 1 + I}, ...]NContourIntegrate[(z ^ 5 + 1/(2z - I)(3z - 1) ^ 3), z∈RegularPolygon[5]]f = ((z - I) ^ 3/(z - 1)(z - 2)(z - 3));reg = Triangle[{{2, 1}, {-2, 0}, {3, -1}}];ComplexPlot[f, {z, -4 - 2I, 4 + 2I}, ...]NContourIntegrate[f, z∈reg]f = (z^4 + 1/(z^2 - 1)^2 + 3);reg = Rectangle[{-2, 0}, {2, 1}];ComplexPlot[f, {z, -5 / 2 - 2I, 5 / 2 + 2I}, ...]NContourIntegrate[f, z∈reg]f = (z ^ 6 - 2I * z ^ 5 + 3 / 2z ^ 3 - 1) / (z(z ^ 2 - z - I) ^ 2);reg = Circle[];ComplexPlot[f, {z, -3 / 2 - 3I / 2, 3 / 2 + 3I / 2}, ...]NContourIntegrate[f, z∈reg]f = ((z + 1) ^ 2(z - 2 - 2I) ^ 3/z^2(z - 1 / 2) ^ 2(z - 3 + I));reg = Line[{{1, 0}, {2, 1}, {3, 0}, {4, 1}}];ComplexPlot[f, {z, -1 - 2I, 9 / 2 + 2I}, ...]NContourIntegrate[f, z∈reg]f = (2z(z + 2)/z - 2);reg = Circle[{0, 0}, {1 / 2, 1}, {0, (7Pi/4)}];ComplexPlot[f, {z, -2 - 2I, 2 + 2I}, ...]NContourIntegrate[f, z∈reg]f = (z - 2) ^ 2 / (z - 1 / 2);reg = Circle[];NContourIntegrate[f, z∈reg]特殊主题：亚纯函数 (5)
f = (z^2 + I/(z - I / 2)Cos[3z]);reg = RegularPolygon[5];ComplexPlot[f, {z, -2 - 2I, 2 + 2I}, ...]NContourIntegrate[f, z∈reg]ContourIntegrate[f, z∈reg]N[%]f = (Sin[z]/z(z^2 + 3)(z - 2));reg = Circle[{0, 0}, {1, 3}];ComplexPlot[f, {z, -4 - 4I, 4 + 4I}, ...]NContourIntegrate[f, z∈reg]f = (Cosh[z^2 - 1 / 4]/z^2 + 1 / 4);reg = Disk[{0, 0}, 1, {0, Pi}];ComplexPlot[f, {z, -2 - 2I, 2 + 2I}, ...]NContourIntegrate[f, z∈reg]f = Tan[z - I];reg = Annulus[{0, 0}, {1, 3}, {0, Pi}];ComplexPlot[f, {z, -7 / 2 - 7I / 2, 7 / 2 + 7I / 2}, ...]NContourIntegrate[f, z∈reg]NContourIntegrate[1 / Sin[z], z∈Circle[{0, 0}, 6]]特殊主题：含有本质奇点的函数 (4)
f = Exp[1 / z];reg = Circle[];ComplexPlot[f, {z, -2 - 2I, 2 + 2I}, ...]NContourIntegrate[f, z∈reg]环线内有本质奇点的 Sin 函数：
f = (Sin[1 / z]/z^4 + I);reg = Rectangle[{-1 / 2, -1 / 2}, {1, 1}];ComplexPlot[f, {z, -2 - 2I, 2 + 2I}, ...]NContourIntegrate[f, z∈reg]f = (Tanh[1 / z]/(z - 1)(z + I / 2));reg = Disk[{-1 / 2, -1}, {1, 3 / 2}, {0, Pi / 2}];ComplexPlot[f, {z, -2 - 2I, 2 + 2I}, ...]NContourIntegrate[f, z∈reg]f = (Cos[2 - 1 / z]/z);reg = Annulus[{-1 / 2, -1}, {1 / 2, 3 / 2}, {0, Pi / 2}];ComplexPlot[f, {z, -2 - 2I, 3 + 2I}, ...]NContourIntegrate[f, z∈reg]特殊主题：非解析函数 (4)
f = Conjugate[z];reg = Circle[];ComplexPlot[f, {z, -2 - 2I, 2 + 2I}, ...]NContourIntegrate[f, z∈reg]Arg 函数的围道积分：
f = Arg[z];reg = Circle[{0, 0}, 2];ComplexPlot[f, {z, -3 - 3I, 3 + 3I}, ...]NContourIntegrate[Arg[z], z∈reg]f = (1/z - Conjugate[z] + 1);reg = Disk[{0, 0}, {1, 2}, {-Pi / 8, 11Pi / 8}];ComplexPlot[f, {z, -2 - 2I, 2 + 2I}, ...]NContourIntegrate[f, z∈reg]//Simplifyf = (Exp[Conjugate[z]]/z + Conjugate[z]);reg = Rectangle[{-5 / 2, -5 / 2}, {-1, -1}];ComplexPlot[f, {z, -6 / 2 - 6 / 2I, 2 + 2I}, ...]NContourIntegrate[f, z∈reg]特殊主题：有分支切割的函数 (2)
f = Sqrt[z ^ 3 - I / 2];reg = Circle[];ComplexPlot[f, {z, -1.5 - 1.5I, 1.5 + 1.5I}, ...]NContourIntegrate[f, z∈reg, WorkingPrecision -> 20, MaxRecursion -> 20, AccuracyGoal -> 6]//Quietf = (z^2/Sqrt[z^2 + 1 / 2]);reg = Circle[];ComplexPlot[f, {z, -2 - 2I, 2 + 2I}, ...]NContourIntegrate[f, z∈reg]//Quiet特殊主题：已命名的环线 (7)
沿实轴正方向、围绕实轴上的极点、闭合于复平面上半部分的围道积分：
NContourIntegrate[I E ^ (I p) / (p ^ 2 - 4), p∈{"UpperSemicircle", 2, -2}]ContourIntegrate[I E ^ (I p) / (p ^ 2 - 4), p∈{"UpperSemicircle", 2, -2}]%//NComplexPlot[I E ^ (I p) / (p ^ 2 - 4), {p, -4 - 4I, 4 + 4I}, ...]NContourIntegrate[I E ^ (I p) / (p ^ 2 - 1) / (p - I / 2), p∈{"UpperSemicircle", Point[{1, 0}], Point[{-1, 0}]}]ContourIntegrate[I E ^ (I p) / (p ^ 2 - 1) / (p - I / 2), p∈{"UpperSemicircle", Point[{1, 0}], Point[{-1, 0}]}]%//N沿实轴正方向、围绕实轴上的极点、闭合于复平面下半部分的围道积分：
NContourIntegrate[I E ^ (-I p) / (p ^ 2 - 4), p∈{"LowerSemicircle", -2, 2}, MaxRecursion -> 20]ContourIntegrate[I E ^ (-I p) / (p ^ 2 - 4), p∈{"LowerSemicircle", -2, 2}]%//NComplexPlot[I E ^ (-I p) / (p ^ 2 - 4), {p, -4 - 4I, 4 + 4I}, ...]NContourIntegrate[I E ^ (-2I p) / (p ^ 2 - 1) / (p + 3I), p∈{"LowerSemicircle", Point[{-1, 0}], Point[{1, 0}]}, MaxRecursion -> 20]ContourIntegrate[I E ^ (-2I p) / (p ^ 2 - 1) / (p + 3I), p∈{"LowerSemicircle", Point[{-1, 0}], Point[{1, 0}]}]%//NNContourIntegrate[(-z) ^ (1 / 2)E ^ (-z), z∈{"Hairpin", HalfLine[{0, 0}, {1, 0}]}]ContourIntegrate[(-z) ^ (1 / 2)E ^ (-z), z∈{"Hairpin", HalfLine[{0, 0}, {1, 0}]}]%//NComplexPlot[(-z) ^ (1 / 2)E ^ (-z), {z, -2 - 2I, 3 + 3I}, ...]NContourIntegrate[(-z) ^ (1 / 3 - 1)E ^ (-z), z∈{"Hairpin", HalfLine[{0, 0}, {1, 0}]}]ContourIntegrate[(-z) ^ (1 / 3 - 1)E ^ (-z), z∈{"Hairpin", HalfLine[{0, 0}, {1, 0}]}]%//NComplexPlot[(-z) ^ (1 / 3 - 1)E ^ (-z), {z, -2 - 2I, 3 + 3I}, ...]计算结果为 Zeta 函数的围道积分：
ContourIntegrate[(-z) ^ (14 / 9 - 1)E ^ (-3 z) / (1 - E ^ -z), z∈{"Hairpin", HalfLine[{0, 0}, {1, 0}]}]%//NNContourIntegrate[(-z) ^ (14 / 9 - 1)E ^ (-3 z) / (1 - E ^ -z), z∈{"Hairpin", HalfLine[{0, 0}, {1, 0}]}]ComplexPlot[(-z) ^ (14 / 9 - 1)E ^ (-3 z) / (1 - E ^ -z), {z, -3 - 3I, 3 + 3I}, ...]NContourIntegrate[E ^ (-z - 1 / 2z ^ 2)(-z) ^ (3 / 2 - 1), z∈{"Hairpin", HalfLine[{0, 0}, {1, 0}]}]ComplexPlot[E ^ (-z - 1 / 2z ^ 2)(-z) ^ (3 / 2 - 1), {z, -3 - 3I, 3 + 3I}, ...]NContourIntegrate[Log[z / (z - 1)], z∈{"Dumbbell", 0, 1}]ComplexPlot[Log[z / (z - 1)], {z, -1 - I, 2 + 2I}, ...]特殊主题：区域环线 (7)
NContourIntegrate[1 / (s + 1)(1 / 2) ^ -s, s∈InfiniteLine[{1, 0}, {0, 1}]]ComplexPlot[1 / (s + 1) * (1 / 2) ^ -s, {s, -20 - 20I, 20 + 20I}, ...]NContourIntegrate[1 / (z - 1 / 2), z∈Circle[]]NContourIntegrate[(z + 1)(z - I)(z + 2I - 1), z∈Line[{{0, 0}, {1, 1}}]]ComplexPlot[(z + 1)(z - I)(z + 2I - 1), {z, -3 - 3I, 3 + 3I}, ...]reg = Triangle[{{0, 1}, {-1, 0}, {1, 0}}];NContourIntegrate[Sinh[z ^ 2] / (z - I / 2), z∈reg]ComplexPlot[Sinh[z ^ 2] / (z - I / 2), {z, -2 - I, 2 + 2I}, ...]reg = Rectangle[{-1, -1}, {1, 1}];NContourIntegrate[E ^ (I z) / (z ^ 2 + 1 / 4), z∈reg]ComplexPlot[E ^ (I z) / (z ^ 2 + 1 / 4), {z, -2 - 2I, 2 + 2I}, ...]reg = Disk[{0, 0}, {1, 2}, {0, Pi / 3}];NContourIntegrate[Cosh[z ^ 3] / (z - 1 / 2 - I), z∈reg]ComplexPlot[Cosh[z ^ 3] / (z - 1 / 2 - I), {z, -1 - I, 2 + 2I}, ...]reg = Annulus[{0, 0}, {1, 2}, {0, Pi}];NContourIntegrate[ArcTan[z + 2] / (z ^ 2 + 3 / 2), z∈reg]ComplexPlot[ArcTan[z + 2] / (z ^ 2 + 3 / 2), {z, -3 - I, 3 + 3I}, ...]选项 (7)
AccuracyGoal (1)
选项 AccuracyGoal 设置准确度的位数：
exact = ContourIntegrate[1 / z, z∈Circle[{2, 0}]]NContourIntegrate[1 / z, z∈Circle[{2, 0}], AccuracyGoal -> 15] - exact默认设置仅设定 PrecisionGoal：
NContourIntegrate[1 / z, z∈Circle[{2, 0}]] - exact
MaxPoints (1)
MaxRecursion (1)
选项 MaxRecursion 指定最大递归步骤数：
NContourIntegrate[(-z) ^ (1 / 2)E ^ (-z), z∈{"Hairpin", HalfLine[{0, 0}, {1, 0}]}, MaxRecursion -> 5]
NContourIntegrate[(-z) ^ (1 / 2)E ^ (-z), z∈{"Hairpin", HalfLine[{0, 0}, {1, 0}]}, MaxRecursion -> 25]ContourIntegrate[(-z) ^ (1 / 2)E ^ (-z), z∈{"Hairpin", HalfLine[{0, 0}, {1, 0}]}]%//NMethod (1)
选项 Method 可接受与 NIntegrate 同样的值. 例如：
NContourIntegrate[(z - 3) / (z - I / 2), z∈Circle[], Method -> "TrapezoidalRule", WorkingPrecision -> 9]NContourIntegrate[(z - 3) / (z - I / 2), z∈Circle[], Method -> "NewtonCotesRule", WorkingPrecision -> 9]NContourIntegrate[(z - 3) / (z - I / 2), z∈Circle[], Method -> "ClenshawCurtisRule", WorkingPrecision -> 9]NContourIntegrate[(z - 3) / (z - I / 2), z∈Circle[], WorkingPrecision -> 9]ContourIntegrate[(z - 3) / (z - I / 2), z∈Circle[]]%//N[#, 9]&MinRecursion (1)
选项 MinRecursion 强制使用最小数量的递归子划分：
NContourIntegrate[Exp[-300Abs[z] ^ 2], z∈Line[{{-70, -70}, {70, 70}}]]
NContourIntegrate[Exp[-300Abs[z] ^ 2], z∈Line[{{-70, -70}, {70, 70}}], MinRecursion -> 5]ContourIntegrate[Exp[-300Abs[z] ^ 2], z∈Line[{{-70, -70}, {70, 70}}]]%//NPrecisionGoal (1)
选项 PrecisionGoal 设置积分中的相对容差：
exact = ContourIntegrate[1 / (1 / 2 + Sin[z]), z∈Circle[]]NContourIntegrate[1 / (1 / 2 + Sin[z]), z∈Circle[], PrecisionGoal -> 15] - exactNContourIntegrate[1 / (1 / 2 + Sin[z]), z∈Circle[]] - exactWorkingPrecision (1)
可用 WorkingPrecision 设置工作精度：
f = Exp[Tan[2 + 1 / z]];NContourIntegrate[f, z∈Circle[]]NContourIntegrate[f, z∈Circle[], WorkingPrecision -> 25]应用 (22)
有理函数 (2)
NContourIntegrate[(1/z^2 + 1), z∈Disk[{0, 0}, 10 ^ 4, {0, Pi}]]
Limit[ContourIntegrate[(1/z^2 + 1), z∈Disk[{0, 0}, R, {0, Pi}]], R -> ∞]用 NIntegrate 可获得同样的结果：
NIntegrate[(1/x^2 + 1), {x, -∞, ∞}]NIntegrate[(x^2/(x^2 + 1)(x^2 + 4)), {x, -∞, ∞}]NContourIntegrate[(z^2/(z^2 + 1)(z^2 + 4)), z∈Disk[{0, 0}, 10 ^ 4, {0, Pi}]]三角函数与有理函数的积 (2)
NIntegrate[(Cos[2x]/(x^2 + 3^2)^2), {x, -∞, ∞}]NIntegrate[(Sin[2x]/(x^2 + 3^2)^2), {x, -∞, ∞}]res = NContourIntegrate[(Exp[2 I z]/(z^2 + 3^2)^2), z∈Disk[{0, 0}, 10 ^ 4, {0, Pi}], Rule[...]]{Re[%], Im[%]}//ChopNIntegrate[(x Sin[2x]/x^2 + 3), {x, -∞, ∞}]NIntegrate[(x Cos[2x]/x^2 + 3), {x, -∞, ∞}]NContourIntegrate[(z Exp[I 2 z]/z^2 + 3), z∈Disk[{0, 0}, 10 ^ 4, {0, Pi}], MaxRecursion -> 20]{Im[%], Re[%]}//Chop三角函数 (3)
f = (1/1 + 1 / 2 * Sin[θ]);NIntegrate[f, {θ, 0, 2Pi}]g = f /. {Sin[θ] -> (z - z^-1/2I)}//FullSimplifyNContourIntegrate[(g/I z), z∈Circle[]]f = (2Cos[2θ]Sin[3θ] ^ 6/1 - 4Cos[θ] + 4);NIntegrate[f, {θ, 0, 2Pi}]g = (f//TrigExpand) /. {Sin[θ] -> (z - z^-1/2I), Cos[θ] -> (z + z^-1/2)}//FullSimplifyNContourIntegrate[(g/I z), z∈Circle[]]f = (1/5 + 4Sin[θ] ^ 7);NIntegrate[f, {θ, 0, 2Pi}]g = f /. {Sin[θ] -> (z - z^-1/2I)}//FullSimplifyNContourIntegrate[(g/I z), z∈Circle[]]傅立叶变换 (2)
ft = FourierTransform[(1/x^2 + 1), x, k]
ft /. k -> 2//N(1/Sqrt[2Pi])NContourIntegrate[(Exp[2I z]/z^2 + 1), z∈Disk[{0, 0}, 10 ^ 4, {0, Pi}], Rule[...]]
ft /. k -> -2//N(-1/Sqrt[2Pi])NContourIntegrate[(Exp[-2I z]/z^2 + 1), z∈Disk[{0, 0}, 10 ^ 4, {-Pi, 0}], MaxRecursion -> 20]ft = FourierTransform[(1/-2 + 2x - x^2), x, k]
ft /. k -> 2//N(1/Sqrt[2Pi])NContourIntegrate[(Exp[2I z]/-2 + 2z - z^2), z∈Disk[{0, 0}, 10 ^ 4, {0, Pi}], Rule[...]]
ft /. k -> -2//N(-1/Sqrt[2Pi])NContourIntegrate[(Exp[-2I z]/-2 + 2z - z^2), z∈Disk[{0, 0}, 10 ^ 3, {-Pi, 0}]]拉普拉斯逆变换 (4)
ilt = InverseLaplaceTransform[1 / (s + 1), s, t, GenerateConditions -> True]
ilt /. t -> 3//N1 / (2Pi I)NContourIntegrate[1 / (s + 1) * E ^ (3s), s∈InfiniteLine[{1, 0}, {0, 1}]]f = Log[(s + 1) / (s + 2)];ilt = InverseLaplaceTransform[f, s, t, GenerateConditions -> True]
ilt /. t -> 2//N1 / (2Pi I)NContourIntegrate[f * E ^ (2s), s∈InfiniteLine[{1, 0}, {0, 1}]]ilt = InverseLaplaceTransform[1 / Sqrt[s + 1], s, t, GenerateConditions -> True]
ilt /. t -> 1 / 2//N1 / (2Pi I)NContourIntegrate[1 / Sqrt[s + 1] * E ^ (s / 2), s∈InfiniteLine[{1, 0}, {0, 1}], MaxRecursion -> 50]含有 Log 的函数的拉普拉斯逆变换：
ilt = InverseLaplaceTransform[Log[s] / (s - 1), s, t, GenerateConditions -> True]
ilt /. t -> 3 / 2//N1 / (2Pi I)NContourIntegrate[Log[s] / (s - 1) * E ^ (s * 3 / 2), s∈InfiniteLine[{1, 0}, {0, 1}], MaxRecursion -> 20]梅林逆变换 (4)
imt = InverseMellinTransform[(1/s^2 - 1), s, x, GenerateConditions -> True]
imt /. x -> 1 / 2//N1 / (2Pi I)NContourIntegrate[(1 / 2) ^ (-s) * (1/s^2 - 1), s∈InfiniteLine[{{2, 0}, {2, 1}}]]imt = InverseMellinTransform[1 / (s - 1), s, x, GenerateConditions -> True]
Assuming[Re[s] > 1, Simplify[imt]] /. x -> 1 / 5//N1 / (2Pi I)NContourIntegrate[1 / (s - 1) * (1 / 5) ^ -s, s∈InfiniteLine[{2, 0}, {0, 1}]]f = (1/(1 + x)^3);MellinTransform[f, x, s, GenerateConditions -> True]
1 / (2Pi I)NContourIntegrate[Gamma[s]Gamma[3 - s] / 2 * (1 / 3) ^ -s, s∈InfiniteLine[{2, 0}, {0, 1}]]f /. x -> 1 / 3//Nf = (x ^ 2/(1 + x)^3);MellinTransform[(x ^ 2/(1 + x)^3), x, s, GenerateConditions -> True]
1 / (2Pi I)NContourIntegrate[(1/2) π s (1 + s) Csc[π s] * (2 / 3) ^ -s, s∈InfiniteLine[{0, 0}, {0, 1}]]f /. x -> 2 / 3//NZ 逆变换 (2)
InverseZTransform[z / (z - 2), z, n]
% /. n -> 3(1/2Pi I)NContourIntegrate[(z^3/(z - 2)), z∈Circle[{0, 0}, 10 ^ 3]]f = InverseZTransform[z / (z ^ 2 - 3 z + 1), z, n]
f /. n -> 7//N(1/2Pi I)NContourIntegrate[z^n / (z ^ 2 - 3 z + 1) /. n -> 7, z∈Circle[{0, 0}, 10 ^ 3], ...]经典定理 (3)
f = 3z ^ 2 / (z ^ 3 - 1 / 2);reg = Circle[];ComplexPlot[f, {z, -2 - 2I, 2 + 2I}, ...]NContourIntegrate[f, z∈reg]
(2Pi I) * ResidueSum[{f, Abs[z] < 1}, z]//N积分环线可以在不改变积分值的情况下改变形状，前提是没有与函数的奇异点相交：
f = 1 / (z - 2) + 1 / (z - 3);reg1 = Line[{{-1, 0}, {0, 1}, {1, 0}}];NContourIntegrate[f, z∈reg1]reg2 = Line[{{-1, 0}, {0, -1}, {1, 0}}];NContourIntegrate[f, z∈reg2]NContourIntegrate[1 / (z - 2), z∈Circle[], AccuracyGoal -> 5]ComplexPlot[1 / (z - 2), {z, -3 - 3I, 3 + 3I}, ...]属性和关系 (6)
如果符号计算失败，可用 N[ContourIntegrate[…]] 获得数值解：
f = Exp[Tan[2 + 1 / z]];ContourIntegrate[f, z∈Circle[]]N[%]也可用 NIntegrate 进行计算：
NIntegrate[I * E ^ (I t) * (f /. z -> E ^ (I t)), {t, 0, 2Pi}]也可用 NContourIntegrate 进行计算：
NContourIntegrate[f, z∈Circle[]]用 NIntegrate 也可以计算数值围道积分：
NContourIntegrate[1 / (z - 1 / 2), z∈Circle[]]NIntegrate[I * E ^ (I t) / (E ^ (I t) - 1 / 2), {t, 0, 2Pi}]NIntegrate 可以沿复平面中的直线进行积分：
NIntegrate[z ^ 2, {z, 0, 1 + I}]NContourIntegrate[z ^ 2, z∈Line[{{0, 0}, {1, 1}}]]也可以使用 ResidueSum 获得闭合路径上的围道积分：
f = z ^ 2 / (2z ^ 3 - 1);NContourIntegrate[f, z∈Circle[{0, 0}, 2]]2πI * ResidueSum[{f, Abs[z] < 2}, z]//N可以使用 FunctionPoles 找到亚纯函数的极点：
f = Cos[z] / (z + 1 / 2);FunctionPoles[f, z]NContourIntegrate[f, z∈Circle[{0, 0}, 1]]也可以使用 Residue 计算积分：
2π I * Residue[f, {z, -1 / 2}]//N也可以使用 Residue 获得闭合路径上的围道积分：
f = z ^ 2 / ((z - 1)(z - 2));NContourIntegrate[f, z∈Circle[{0, 0}, 3]]2π I * (Residue[f, {z, 1}] + Residue[f, {z, 2}])//N互动范例 (2)
f = (1/z^2 + 1);Manipulate[
ComplexPlot[f, {z, -6 - 6I, 6 + 6I}, ...]
, {R, 0.1, 7}, SaveDefinitions -> True]Manipulate[NContourIntegrate[f, z∈Disk[{0, 0}, R, {0, Pi}]], {R, 0.1, 7}, SaveDefinitions -> True]与 ContourIntegrate 相比较：
ContourIntegrate[f, z∈Disk[{0, 0}, R, {0, Pi}]]f = (1/z^6 + 1);Manipulate[
ComplexPlot[f, {z, -6 - 6I, 6 + 6I}, ...], {R, 0.1, 7}, SaveDefinitions -> True]Manipulate[NContourIntegrate[f, z∈Disk[{0, 0}, R, {0, Pi}]], {R, 0.1, 7}, SaveDefinitions -> True]与 ContourIntegrate 相比较：
ContourIntegrate[f, z∈Disk[{0, 0}, R, {0, Pi}]]
Wolfram Research (2024)，NContourIntegrate，Wolfram 语言函数，https://reference.wolfram.com/language/ref/NContourIntegrate.html.
文本
Wolfram Research (2024)，NContourIntegrate，Wolfram 语言函数，https://reference.wolfram.com/language/ref/NContourIntegrate.html.
CMS
Wolfram 语言. 2024. "NContourIntegrate." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/NContourIntegrate.html.
APA
Wolfram 语言. (2024). NContourIntegrate. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/NContourIntegrate.html 年