📚 Further Core Pure 2 Revision: Complex Numbers, Matrices and More | 进阶核心纯数2复习：复数、矩阵与更多考纲要点
Further Core Pure 2 extends the ideas met in Further Core Pure 1 and introduces more advanced tools for pure mathematics. This unit focuses on complex numbers in modulus-argument form, De Moivre’s theorem, hyperbolic functions, eigenvalues and eigenvectors, matrix diagonalisation, vectors and planes, polar coordinates, series, and second-order differential equations. The key to success is not just remembering formulas, but understanding how the topics connect and how to select the most efficient method in an exam.
进阶核心纯数2在进阶核心纯数1的基础上继续延伸，引入更高级的纯数学工具。本单元重点包括复数的模-辐角形式、棣莫弗定理、双曲函数、特征值与特征向量、矩阵对角化、向量与平面、极坐标、级数以及二阶微分方程。取得高分的关键不只是记住公式，而是理解各主题之间的联系，并在考试中能够选择最高效的方法。
1. Complex Numbers and De Moivre’s Theorem | 复数与棣莫弗定理
Any complex number z = x + iy can be written in modulus-argument form as z = r (cos θ + i sin θ), where r = |z| = √(x² + y²) and θ = arg z. Multiplication and division become much simpler in this form: multiply or divide the moduli, and add or subtract the arguments.
任何复数 z = x + iy 都可以写成模-辐角形式 z = r (cos θ + i sin θ)，其中 r = |z| = √(x² + y²)，θ = arg z。在这种形式下，乘法和除法会变得简单很多：模相乘或相除，辐角相加或相减。
z₁z₂ = r₁r₂ (cos(θ₁ + θ₂) + i sin(θ₁ + θ₂))
De Moivre’s theorem states that for any integer n, (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. This is the standard method for finding powers of a complex number and also for deriving multiple-angle trigonometric identities.
棣莫弗定理指出，对于任意整数 n，有 (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。这是求复数幂的标准方法，也用于推导多倍角三角恒等式。
zⁿ = rⁿ (cos nθ + i sin nθ)
A typical exam question asks you to express cos 3θ in terms of cos θ, or to evaluate (1 + i)¹⁰ exactly. Start by writing the base in modulus-argument form, apply De Moivre’s theorem, then convert back to Cartesian form if required.
常见的考题会要求用 cos θ 表示 cos 3θ，或者精确计算 (1 + i)¹⁰。解题时应先将底数写成模-辐角形式，应用棣莫弗定理，然后在需要时再转换回笛卡尔形式。
2. Roots of Unity and Complex Loci | 单位根与复数轨迹
The nth roots of unity are the solutions of zⁿ = 1. They are given by z = cos(2kπ/n) + i sin(2kπ/n) for k = 0, 1, 2, …, n−1, and they lie evenly spaced on the unit circle in the Argand diagram.
n 次单位根是方程 zⁿ = 1 的解，可表示为 z = cos(2kπ/n) + i sin(2kπ/n)，其中 k = 0, 1, 2, …, n−1。它们在阿甘特图中均匀分布在单位圆上。
Roots of a general complex number w are found by writing w = r (cos θ + i sin θ), then using z = r^(1/n) [cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)]. The principal root is usually the one with k = 0, but all n roots must be given unless the question states otherwise.
一般复数 w 的 n 次方根可以通过将 w 写成 w = r (cos θ + i sin θ)，再使用 z = r^(1/n) [cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] 来求得。通常主根取 k = 0，但除非题目另有说明，否则应给出全部 n 个根。
Complex loci questions require you to interpret conditions such as |z − a| = r, which is a circle centre a and radius r, or |z − a| = |z − b|, which is the perpendicular bisector of the segment joining a and b. Shading inequalities like |z − 2| < 3 means the inside of a circle, and arg(z − 1) = π/4 gives a half-line.
复数轨迹问题要求解释诸如 |z − a| = r（以 a 为圆心、r 为半径的圆）或 |z − a| = |z − b|（连接 a 和 b 线段的垂直平分线）等条件。对不等式如 |z − 2| < 3 进行着色表示圆内部，而 arg(z − 1) = π/4 表示一条半直线。
3. Hyperbolic Functions | 双曲函数
The hyperbolic functions are defined in terms of exponentials: cosh x = (eˣ + e⁻ˣ)/2 and sinh x = (eˣ − e⁻ˣ)/2. They appear in integration, differential equations, and many applied contexts.
双曲函数通过指数函数定义：cosh x = (eˣ + e⁻ˣ)/2，sinh x = (eˣ − e⁻ˣ)/2。它们出现在积分、微分方程以及许多应用背景中。
The fundamental identity is cosh² x − sinh² x = 1. Notice the minus sign, which is the key difference from the trigonometric identity cos² x + sin² x = 1. Other useful relations include sinh 2x = 2 sinh x cosh x and cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 2 sinh² x + 1.
基本恒等式是 cosh² x − sinh² x = 1。注意这里是负号，这是与三角恒等式 cos² x + sin² x = 1 的关键区别。其他常用关系包括 sinh 2x = 2 sinh x cosh x 以及 cosh 2x = cosh² x + sinh² x = 2 cosh² x − 1 = 2 sinh² x + 1。
Osborn’s rule helps convert trigonometric identities into hyperbolic identities: replace cos by cosh, sin by i sinh, and change the sign of any product of two sinh terms. This is useful when you need a hyperbolic identity but have only memorised the trigonometric version.
奥斯本法则可帮助将三角恒等式转换为双曲恒等式：将 cos 替换为 cosh，将 sin 替换为 i sinh，并改变任何两个 sinh 乘积项的符号。当你只记住了三角版本时，这一方法非常有用。
4. Inverse Hyperbolic Functions and Logarithmic Forms | 反双曲函数与对数形式
The inverse hyperbolic functions arsinh x, arcosh x and artanh x can all be expressed in terms of natural logarithms. These logarithmic forms are particularly important for integration and for solving equations involving hyperbolic functions.
反双曲函数 arsinh x、arcosh x 和 artanh x 都可以用自然对数表示。这些对数形式在积分和求解含有双曲函数的方程时尤为重要。
arsinh x = ln(x + √(x² + 1))
arcosh x = ln(x + √(x² − 1)), x ≥ 1
artanh x = ½ ln((1 + x)/(1 − x)), |x| < 1
The derivatives of inverse hyperbolic functions are worth remembering: d/dx arsinh x = 1/√(x² + 1), d/dx arcosh x = 1/√(x² − 1), and d/dx artanh x = 1/(1 − x²). These lead directly to standard integrals such as ∫ 1/√(x² + a²) dx = arsinh(x/a) + c.
反双曲函数的导数值得记住：d/dx arsinh x = 1/√(x² + 1)，d/dx arcosh x = 1/√(x² − 1)，d/dx artanh x = 1/(1 − x²)。这些导数直接对应标准积分，例如 ∫ 1/√(x² + a²) dx = arsinh(x/a) + c。
5. Matrices: Eigenvalues and Eigenvectors | 矩阵：特征值与特征向量
For an n × n matrix A, a non-zero vector v is an eigenvector if A v = λ v for some scalar λ. The scalar λ is the eigenvalue associated with v. Eigenvectors indicate directions that are left unchanged by the transformation, apart from being scaled by λ.
对于 n × n 矩阵 A，如果非零向量 v 满足 A v = λ v，其中 λ 为某个标量，则称 v 为特征向量。标量 λ 是与 v 对应的特征值。特征向量表示在变换下方向保持不变、仅被 λ 缩放的向量。
To find eigenvalues, solve the characteristic equation det(A − λ I) = 0. For a 2 × 2 matrix this gives a quadratic equation in λ. Once an eigenvalue is found, substitute it back into (A − λ I) v = 0 to find the corresponding eigenvector.
求特征值需要解特征方程 det(A − λ I) = 0。对于 2 × 2 矩阵，这会得到一个关于 λ 的二次方程。求出特征值后，将其代回 (A − λ I) v = 0 即可求得对应的特征向量。
For a 2 × 2 matrix A = [[a, b], [c, d]], the characteristic equation can be written as λ² − tr(A) λ + det(A) = 0, where tr(A) = a + d and det(A) = ad − bc. This is often quicker than expanding the determinant from scratch.
对于 2 × 2 矩阵 A = [[a, b], [c, d]]，特征方程可写成 λ² − tr(A) λ + det(A) = 0，其中 tr(A) = a + d，det(A) = ad − bc。这通常比从头展开行列式更快。
6. Matrix Diagonalisation and Powers of Matrices | 矩阵对角化与矩阵的幂
If a matrix A has distinct linearly independent eigenvectors, it can be diagonalised as A = P D P⁻¹, where D is a diagonal matrix containing the eigenvalues, and P is the matrix whose columns are the corresponding eigenvectors.
如果矩阵 A 有线性无关的特征向量，就可以将其对角化为 A = P D P⁻¹，其中 D 是包含特征值的对角矩阵，P 的列是对应的特征向量。
The main use of diagonalisation is to compute high powers of a matrix efficiently. Since Aⁿ = P Dⁿ P⁻¹, and Dⁿ is easy to find by simply raising each diagonal entry to the power n, this avoids repeated matrix multiplication.
对角化的主要用途是高效计算矩阵的高次幂。由于 Aⁿ = P Dⁿ P⁻¹，而 Dⁿ 只需将每个对角元素分别求 n 次幂即可，因此避免了重复的矩阵乘法。
Remember that P⁻¹ must be found carefully. For a 2 × 2 matrix P = [[p, q], [r, s]], the inverse is P⁻¹ = 1/(ps − qr) [[s, −q], [−r, p]]. A common exam mistake is to write A = P⁻¹ D P instead of A = P D P⁻¹, so check the order.
务必仔细求 P⁻¹。对于 2 × 2 矩阵 P = [[p, q], [r, s]]，其逆矩阵为 P⁻¹ = 1/(ps − qr) [[s, −q], [−r, p]]。考试中常见错误是把 A = P D P⁻¹ 写成 A = P⁻¹ D P，因此要检查顺序。
7. Further Vectors: Planes and Intersections | 进阶向量：平面与相交问题
In three dimensions, a plane can be described in vector form as r · n = a · n, where n is a normal vector to the plane and a is the position vector of a point on the plane. The Cartesian equation is n₁ x + n₂ y + n₃ z = d.
在三维空间中，平面可用向量形式 r · n = a · n 描述，其中 n 是平面的法向量，a 是平面上一点的位置向量。其笛卡尔方程为 n₁ x + n₂ y + n₃ z = d。
To find the line of intersection of two planes, solve their equations simultaneously. The direction of the line is the cross product of the two normal vectors, and one point on the line can be found by setting one coordinate equal to zero and solving the remaining two equations.
求两个平面的交线时，需要联立求解它们的方程。交线的方向是两个法向量的叉积，交线上的一点可以令某个坐标为零，再解剩余两个方程得到。
The angle between two planes is the angle between their normal vectors, found using the dot product: cos θ = (n₁ · n₂) / (|n₁||n₂|). The distance from a point with position vector p to the plane r · n = d is |p · n − d| / |n|.
两个平面之间的夹角就是它们法向量之间的夹角，可用点积求得：cos θ = (n₁ · n₂) / (|n₁||n₂|)。位置向量为 p 的点到平面 r · n = d 的距离为 |p · n − d| / |n|。
8. Polar Coordinates | 极坐标
Polar coordinates describe a point P by the distance r from the origin and the angle θ from the initial line, with r ≥ 0 and θ measured in radians. The conversion formulas are x = r cos θ and y = r sin θ.
极坐标通过到原点的距离 r 和从极轴起的角度 θ 来描述点 P，其中 r ≥ 0，θ 以弧度为单位。转换公式为 x = r cos θ 和 y = r sin θ。
The area enclosed by a polar curve r = f(θ) between θ = α and θ = β is given by A = ½ ∫ₐ^β r² dθ. You must make sure the limits are in radians and the curve does not overlap the region unintentionally.
极坐标曲线 r = f(θ) 在 θ = α 到 θ = β 之间围成的面积由 A = ½ ∫ₐ^β r² dθ 给出。必须确保积分限以弧度为单位，并且曲线没有意外重叠区域。
To find tangents to a polar curve that are parallel or perpendicular to the initial line, write x = r cos θ and y = r sin θ, then use dy/dx = (dy/dθ) / (dx/dθ). For a tangent parallel to the initial line, set dy/dθ = 0; for a tangent perpendicular to the initial line, set dx/dθ = 0.
求极坐标曲线的平行或垂直于极轴的切线时，先将 x = r cos θ、y = r sin θ，再使用 dy/dx = (dy/dθ) / (dx/dθ)。平行于极轴的切线令 dy/dθ = 0；垂直于极轴的切线令 dx/dθ = 0。
9. Series and the Method of Differences | 级数与差分法
The standard summation results for natural numbers are: ∑ r = n(n+1)/2, ∑ r² = n(n+1)(2n+1)/6, and ∑ r³ = n²(n+1)²/4. These are used to evaluate polynomial series directly.
自然数的标准求和结果包括：∑ r = n(n+1)/2，∑ r² = n(n+1)(2n+1)/6，∑ r³ = n²(n+1)²/4。这些公式用于直接计算多项式级数。
The method of differences is used for series whose terms can be split into two parts that mostly cancel. A common form is to rewrite a term f(r) as g(r) − g(r+1), so that when summed from r = 1 to n, only the first and last parts remain.
差分法适用于可将项拆分为两部分并大部分相互抵消的级数。常见形式是将项 f(r) 改写为 g(r) − g(r+1)，这样从 r = 1 到 n 求和后，只剩下首尾部分。
For example, to evaluate ∑ 1/(r(r+1)), write 1/(r(r+1)) = 1/r − 1/(r+1). The sum becomes (1 − 1/2) + (1/2 − 1/3) + … + (1/n − 1/(n+1)) = 1 − 1/(n+1).
例如，计算 ∑ 1/(r(r+1)) 时，可写成 1/(r(r+1)) = 1/r − 1/(r+1)。求和后得到 (1 − 1/2) + (1/2 − 1/3) + … + (1/n − 1/(n+1)) = 1 − 1/(n+1)。
10. Second-Order Differential Equations | 二阶微分方程
A linear second-order differential equation with constant coefficients has the form a y” + b y’ + c y = f(x), where a, b, c are constants. The solution is made up of the complementary function y_c and a particular integral y_p.
常系数线性二阶微分方程的形式为 a y” + b y’ + c y = f(x)，其中 a、b、c 为常数。其解由补函数 y_c 和特解 y_p 组成。
To find the complementary function, solve the auxiliary equation a m² + b m + c = 0. If the roots are real and distinct, y_c = A e^(m₁x) + B e^(m₂x); if equal, y_c = (A + Bx) e^(mx); if complex m = p ± iq, then y_c = e^(px) (A cos qx + B sin qx).
求补函数需解辅助方程 a m² + b m + c = 0。若根为不同实数，则 y_c = A e^(m₁x) + B e^(m₂x)；若根相等，则 y_c = (A + Bx) e^(mx)；若根为复数 m = p ± iq，则 y_c = e^(px) (A cos qx + B sin qx)。
The particular integral depends on f(x). For f(x) = k, try y = C; for f(x) = k e^(px), try y = C e^(px); for f(x) = k cos px or k sin px, try y = C cos px + D sin px. Substitute into the differential equation to determine the constants.
特解的形式取决于 f(x)。当 f(x) = k 时，试 y = C；当 f(x) = k e^(px) 时，试 y = C e^(px)；当 f(x) = k cos px 或 k sin px 时，试 y = C cos px + D sin px。代入微分方程即可确定常数。
The general solution is y = y_c + y_p. If initial conditions are given, use them to find the arbitrary constants A and B, but only after writing down the full general solution.
通解为 y = y_c + y_p。若给出初始条件，应在写出完整通解之后再利用它们求出任意常数 A 和 B。
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