AS Further Maths: Formula Summary Handbook | AS 进阶数学：公式汇总手册

📚 AS Further Maths: Formula Summary Handbook | AS 进阶数学：公式汇总手册

Welcome to the AS Further Maths Formula Summary Handbook. This guide compiles the essential formulas needed for AS-level Further Mathematics, covering complex numbers, matrices, vectors, series, hyperbolic functions, roots of polynomials, polar coordinates, further calculus, differential equations, and proof by induction. Each section is presented with clear formulas and bilingual explanations to help you revise efficiently.

欢迎使用 AS 进阶数学公式汇总手册。本手册整理了 AS 进阶数学所需的核心公式，涵盖复数、矩阵、向量、级数、双曲函数、多项式根、极坐标、进阶微积分、微分方程和归纳法证明。每个部分都配有清晰的公式和中英双语解释，帮助你高效复习。

1. Complex Numbers | 复数

A complex number is written as z = x + iy, where x, y ∈ ℝ and i = √−1. The modulus is |z| = √(x² + y²), and the argument arg(z) = θ satisfies tan θ = y/x. The complex conjugate is z̅ = x − iy.

复数写作 z = x + iy，其中 x, y ∈ ℝ，i = √−1。模长为 |z| = √(x² + y²)，辐角 arg(z) = θ 满足 tan θ = y/x。共轭复数为 z̅ = x − iy。

Key identities: de Moivre’s theorem states (r(cosθ + i sinθ))ⁿ = rⁿ(cos nθ + i sin nθ). Euler’s formula is eⁱᶿ = cosθ + i sinθ. The nth roots of unity are given by z = cos(2kπ/n) + i sin(2kπ/n) for k = 0,1,…,n−1.

核心恒等式：棣莫弗定理 (r(cosθ + i sinθ))ⁿ = rⁿ(cos nθ + i sin nθ)。欧拉公式 eⁱᶿ = cosθ + i sinθ。单位根的 n 次方根为 z = cos(2kπ/n) + i sin(2kπ/n), k = 0,1,…,n−1。

2. Matrices | 矩阵

For a 2×2 matrix A = [a b; c d], the determinant is det(A) = ad − bc. If det(A) ≠ 0, the inverse is A⁻¹ = 1/(ad−bc) [d −b; −c a]. Matrix multiplication AB is defined only if the number of columns of A equals the number of rows of B.

对于 2×2 矩阵 A = [a b; c d]，行列式为 det(A) = ad − bc。若 det(A) ≠ 0，逆矩阵为 A⁻¹ = 1/(ad−bc) [d −b; −c a]。矩阵乘法 AB 仅当 A 的列数等于 B 的行数时才有定义。

To solve a linear system Ax = b, if A is invertible the solution is x = A⁻¹b. The zero matrix and identity matrix I play roles analogous to 0 and 1 in arithmetic.

求解线性方程组 Ax = b 时，若 A 可逆，解为 x = A⁻¹b。零矩阵与单位矩阵 I 的作用类似于算术中的 0 与 1。

3. Vectors | 向量

Dot product: a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃. It measures the projection of one vector onto another. Cross product (in ℝ³): a×b = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k. The magnitude |a×b| equals the area of the parallelogram spanned by a and b.

点积：a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃，它度量一个向量在另一个向量上的投影。叉积（三维）：a×b = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k。其大小 |a×b| 等于 a 和 b 张成的平行四边形面积。

Vector equation of a line: r = a + λd, where a is a point on the line and d is the direction vector. Plane equation: r·n = d, with n being the normal vector. The scalar triple

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