OCR Year 13 Further Maths Formula & Theorem Quick Reference | OCR 13年级进阶数学公式定理速查手册

📚 OCR Year 13 Further Maths Formula & Theorem Quick Reference | OCR 13年级进阶数学公式定理速查手册

This quick reference guide compiles essential formulae and theorems from the OCR Year 13 Further Mathematics syllabus, covering Pure Core topics such as complex numbers, matrices, vectors, hyperbolic functions, differential equations, polar coordinates, series expansions, and proof by induction. Use it for revision and exam preparation.

本速查手册汇编了OCR 13年级进阶数学教学大纲中的核心公式与定理，涵盖复数、矩阵、向量、双曲函数、微分方程、极坐标、级数展开以及归纳法证明等纯数内容，适用于复习和备考。

1. Complex Numbers & De Moivre’s Theorem | 复数与棣莫弗定理

A complex number can be expressed in polar form z = r(cos θ + i sin θ) with modulus |z| = r and argument arg(z) = θ. The exponential form is given by Euler’s formula, eⁱθ = cos θ + i sin θ, so z = r eⁱθ.

z = r(cos θ + i sin θ) = r eⁱθ

复数可以用极坐标形式表示，模为 |z| = r，辐角为 arg(z) = θ。欧拉公式给出了指数形式，即 z = r eⁱθ。

De Moivre’s theorem states that for any integer n: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. This extends to powers: zⁿ = rⁿ eⁱⁿθ = rⁿ(cos nθ + i sin nθ).

(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ

Published by TutorHao | Year 13 进阶数学 Revision Series | aleveler.com

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