A-Level Edexcel Further Mathematics: Mind Map for Quick Memorization | A-Level Edexcel 进阶数学：思维导图速记

📚 A-Level Edexcel Further Mathematics: Mind Map for Quick Memorization | A-Level Edexcel 进阶数学：思维导图速记

This mind map distils the essential formula sheets and conceptual links for the Edexcel A-Level Further Mathematics syllabus. It is designed to help you visualise connections across Core Pure topics and to speed up last-minute revision. Each branch condenses key definitions, theorems, and standard results that you must recall in the exam.

这份思维导图提炼了 Edexcel A-Level 进阶数学大纲中的所有核心公式与概念关联，旨在帮助你可视化各纯数主题之间的脉络，并在考前快速复习。每个分支都浓缩了必须记住的关键定义、定理和标准结果。

1. Complex Numbers: Basics & De Moivre | 复数基础与棣莫弗定理

A complex number can be written in Cartesian form z = x + iy, where x, y ∈ ℝ and i² = −1. Its modulus is |z| = √(x² + y²) and its argument is arg(z) = θ, usually chosen in (−π, π].

复数可写作直角坐标形式 z = x + iy，其中 x, y 为实数，i² = −1。模为 |z| = √(x² + y²)，辐角为 arg(z) = θ，通常取主值范围 (−π, π]。

The exponential form z = re^(iθ) links trigonometry and algebra. De Moivre’s theorem states (cosθ + i sinθ)^n = cos nθ + i sin nθ for integer n. This is used to find powers and n-th roots of complex numbers.

指数形式 z = re^(iθ) 将三角与代数联系起来。棣莫弗定理指出 (cosθ + i sinθ)^n = cos nθ + i sin nθ（n 为整数），用于求复数的乘幂与 n 次方根。

To solve zⁿ = a, write a in modulus-argument form, then the n distinct roots are given by z_k = r^(1/n) [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)], k = 0, 1, …, n−1. The roots lie on a circle of radius r^(1/n) at equally spaced angles.

解方程 zⁿ = a 时，先将 a 写为模-辐角形式，则 n 个互异根为 z_k = r^(1/n) [cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)]，k = 0,1,…,n−1。这些根位于半径为 r^(1/n) 的圆上，且等角度分布。

Trigonometric identities can be derived by expanding (cosθ + i sinθ)^n using De Moivre and equating real and imaginary parts. For example, cos 3θ = 4 cos³θ − 3 cosθ.

利用棣莫弗定理展开 (cosθ + i sinθ)^n 并取实部和虚部，可推导出三角恒等式，如 cos 3θ = 4 cos³θ − 3 cosθ。

2. Complex Numbers: Regions & Transformations | 复数区域与变换

An inequality such as |z − (a+ib)| < r describes the interior of a

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