📚 AS OCR Further Maths: Quick Reference Formula & Theorem Handbook | AS OCR 进阶数学：公式定理速查手册
This handbook provides a concise reference of key formulas and theorems required for AS-Level OCR Further Mathematics. Use it alongside your revision to strengthen recall and problem‑solving across pure and applied topics. Each section presents a formula or concept in English, immediately followed by its Chinese translation, ensuring clarity for bilingual learners.
本速查手册汇集了 AS 阶段 OCR 进阶数学所需的核心公式与定理，旨在辅助复习，巩固记忆，提升解题能力。每个条目均采用英文原文与中文译文逐条配对呈现，方便双语学习者准确理解。
1. Complex Numbers | 复数
A complex number is written in Cartesian form as z = a + b i, where a, b ∈ ℝ and i² = −1. The real part is Re(z) = a, the imaginary part Im(z) = b.
复数可用笛卡尔形式表示为 z = a + b i，其中 a, b ∈ ℝ 且 i² = −1。实部为 Re(z) = a，虚部为 Im(z) = b。
The modulus is r = |z| = √(a² + b²), and the argument θ = arg(z) satisfies tan θ = b/a, with the quadrant taken into account. The principal argument normally lies in (−π, π].
模为 r = |z| = √(a² + b²)，辐角 θ = arg(z) 满足 tan θ = b/a，并需根据象限确定。主辐角通常取值在 (−π, π] 范围内。
Polar form: z = r (cos θ + i sin θ). Euler’s relation gives the exponential form z = r e^(iθ).
极坐标形式：z = r (cos θ + i sin θ)。利用欧拉公式可得指数形式 z = r e^(iθ)。
Multiplication and division in polar form: z₁ z₂ = r₁ r₂ e^(i(θ₁+θ₂)), z₁ / z₂ = (r₁ / r₂) e^(i(θ₁−θ₂)).
极坐标下的乘除运算：z₁ z₂ = r₁ r₂ e^(i(θ₁+θ₂))，z₁ / z₂ = (r₁ / r₂) e^(i(θ₁−θ₂))。
De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ for any integer n. This is used to find powers and roots of complex numbers.
棣莫弗定理：对于任意整数 n，(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。常用于求复数的幂和根。
The n nth roots of z = r e^(iθ) are given by z^(1/n) = r^(1/n) [ cos((θ+2kπ)/n) + i sin((θ+2kπ)/n) ], k = 0,1,…, n−1.
复数 z = r e^(iθ) 的 n 个 n 次方根为 z^(1/n) = r^(1/n) [ cos((θ+2kπ)/n) + i sin((θ+2kπ)/n) ]，其中 k = 0,1,…, n−1。
2. Matrices and Determinants | 矩阵与行列式
For a 2×2 matrix M = [[a, b], [c, d]], the determinant is det(M) = ad − bc. M is singular if det(M) = 0; otherwise it is non‑singular and its inverse is M⁻¹ = (1/det(M)) [[d, −b], [−c, a]].
对于 2×2 矩阵 M = [[a, b], [c, d]]，行列式为 det(M) = ad − bc。若 det(M) = 0，矩阵是奇异的；否则是非奇异的，其逆矩阵为 M⁻¹ = (1/det(M)) [[d, −b], [−c, a]]。
The transpose Mᵀ is obtained by swapping rows and columns. (M⁻¹)ᵀ = (Mᵀ)⁻¹.
转置矩阵 Mᵀ 通过交换行与列得到。满足 (M⁻¹)ᵀ = (Mᵀ)⁻¹。
For a 3×3 matrix, the determinant can be expanded along any row or column, e.g. det(A) = a₁₁ (a₂₂ a₃₃ − a₂₃ a₃₂) − a₁₂ … + …
对于 3×3 矩阵，行列式可按任一行或列展开，如 det(A) = a₁₁ (a₂₂ a₃₃ − a₂₃ a₃₂) − a₁₂ … + …
Inverse of a 3×3 matrix: A⁻¹ = (1/det(A)) adj(A), where adj(A) is the adjugate (transpose of the cofactor matrix).
3×3 矩阵的逆矩阵：A⁻¹ = (1/det(A)) adj(A)，其中 adj(A) 为伴随矩阵（余子式矩阵的转置）。
A linear transformation in the plane can be represented by a matrix. The image of a point (x, y) is given by multiplication: [[a, b], [c, d]] (x, y)ᵀ = (ax+by, cx+dy)ᵀ.
平面上的线性变换可用矩阵表示。点 (x, y) 的像通过矩阵乘法求得：[[a, b], [c, d]] (x, y)ᵀ = (ax+by, cx+dy)ᵀ。
The determinant of a transformation matrix gives the area scale factor. A negative determinant indicates a reflection combined with rotation.
变换矩阵的行列式给出面积缩放因子。行列式为负表示变换包含反射与旋转。
3. Roots of Polynomials | 多项式根的关系
For a quadratic ax² + bx + c = 0 with roots α and β: α + β = −b/a, αβ = c/a.
对于一元二次方程 ax² + bx + c = 0，两根 α, β 满足：α + β = −b/a，αβ = c/a。
For a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ: Σα = −b/a, Σαβ = c/a, αβγ = −d/a.
三次方程 ax³ + bx² + cx + d = 0 的三根 α, β, γ 满足：Σα = −b/a，Σαβ = c/a，αβγ = −d/a。
For a quartic ax⁴ + bx³ + cx² + dx + e = 0 with roots α, β, γ, δ: Σα = −b/a, Σαβ = c/a, Σαβγ = −d/a, αβγδ = e/a.
四次方程 ax⁴ + bx³ + cx² + dx + e = 0 的四根关系为：Σα = −b/a，Σαβ = c/a，Σαβγ = −d/a，αβγδ = e/a。
Symmetric functions of roots, such as α²+β² = (α+β)² − 2αβ, are often required. These can be expressed using the sums and products of roots without solving the equation directly.
根的对称函数（如 α²+β² = (α+β)² − 2αβ）常需计算，可利用根的和与积直接表示，无需解出具体根。
These relationships allow the construction of new equations whose roots are related to the original ones (e.g., roots squared or reciprocal).
利用这些关系，可构造与原方程根相关的新方程（例如以原根的平方或倒数作为新根）。
4. Summation of Series | 级数求和
Standard results for sums of integers: Σ_{r=1}^{n} r = n(n+1)/2, Σ r² = n(n+1)(2n+1)/6, Σ r³ = n²(n+1)²/4.
整数的求和标准结果：Σ_{r=1}^{n} r = n(n+1)/2，Σ r² = n(n+1)(2n+1)/6，Σ r³ = n²(n+1)²/4。
The method of differences is useful when terms can be written as f(r) − f(r+1) or similar, leading to cancellation.
差分法适用于将通项表示为 f(r) − f(r+1) 等形式，从而通过错位相消求和。
For example, Σ 1/(r(r+1)) = 1 − 1/(n+1) using partial fractions and the method of differences.
例如 Σ 1/(r(r+1)) 可先部分分式分解为 1/r − 1/(r+1)，再用差分法得结果 1 − 1/(n+1)。
Proof by induction is often required to verify sum formulas. Assume true for n = k, then prove for n = k+1.
常需用数学归纳法证明求和公式。假设 n = k 时成立，然后证明 n = k+1 时成立。
5. Binomial Expansion | 二项式展开
For a positive integer n, (1+x)ⁿ = 1 + n x + n(n-1)/2! x² + … + n!/(r!(n−r)!) xʳ + … + xⁿ. The expansion is finite and valid for all x.
对于正整数 n，(1+x)ⁿ = 1 + n x + n(n-1)/2! x² + … + n!/(r!(n−r)!) xʳ + … + xⁿ。该展开式为有限项，对任意 x 均成立。
For (a+b)ⁿ, use (a+b)ⁿ = aⁿ (1 + b/a)ⁿ and expand.
对于 (a+b)ⁿ，可化为 aⁿ (1 + b/a)ⁿ 后再展开。
When n is not a positive integer (e.g., n = −1, 1/2, …), the binomial expansion (1+x)ⁿ = 1 + n x + n(n-1)/2! x² + … is an infinite series valid for |x| < 1.
当 n 不是正整数（如 n = −1, 1/2 等），(1+x)ⁿ = 1 + n x + n(n-1)/2! x² + … 为无穷级数，收敛域为 |x| < 1。
The expansion of (a+bx)ⁿ for non‑integer n requires rewriting as aⁿ (1 + (b/a)x)ⁿ and expanding in powers of x, valid for |(b/a)x| < 1.
对于非整数指数 n 的 (a+bx)ⁿ，需写为 aⁿ (1 + (b/a)x)ⁿ 再展开为 x 的幂级数，要求 |(b/a)x| < 1。
Terms are often left in forms like x² × n(n-1)/2, but simplification and approximation for small x may be required.
各项通常保留形式如 x² × n(n-1)/2，但有时需化简或对小量 x 进行近似。
6. Vectors | 向量
A vector in 3D is written as v = a i + b j + c k or as a column vector. Magnitude: |v| = √(a² + b² + c²).
三维向量记为 v = a i + b j + c k 或列向量形式。模长为 |v| = √(a² + b² + c²)。
Scalar (dot) product: a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃. Used to find angles and to test perpendicularity (a · b = 0).
数量积（点积）：a · b = |a||b| cos θ = a₁b₁ + a₂b₂ + a₃b₃。用于求夹角及判断垂直（a · b = 0）。
The vector (cross) product a × b returns a vector perpendicular to both: magnitude |a×b| = |a||b| sin θ, direction given by the right‑hand rule. In component form, a × b = (a₂b₃ − a₃b₂) i + (a₃b₁ − a₁b₃) j + (a₁b₂ − a₂b₁) k.
向量积（叉积）a × b 结果为垂直于 a 和 b 的向量：大小 |a×b| = |a||b| sin θ，方向由右手定则确定。分量形式：a × b = (a₂b₃ − a₃b₂) i + (a₃b₁ − a₁b₃) j + (a₁b₂ − a₂b₁) k。
Vector equation of a line: r = a + λ d, where a is a point on the line, d is the direction vector, and λ is a parameter.
直线的向量方程：r = a + λ d，其中 a 为线上一点，d 为方向向量，λ 为参数。
Equation of a plane: r · n = p, or r = a + λ u + μ v, where n is normal to the plane, and u, v are direction vectors in the plane.
平面的向量方程：r · n = p 或 r = a + λ u + μ v，其中 n 为法向量，u, v 为平面内两个方向向量。
Intersection problems: substitute line equation into plane equation to solve for λ, or solve simultaneous equations for lines/planes.
求交点的思路：将直线参数方程代入平面方程解 λ，或联立方程求解线与线、面与面的关系。
7. Trigonometry | 三角学
Key identities: sin²θ + cos²θ ≡ 1, 1 + tan²θ ≡ sec²θ, 1 + cot²θ ≡ csc²θ.
基本恒等式：sin²θ + cos²θ ≡ 1，1 + tan²θ ≡ sec²θ，1 + cot²θ ≡ csc²θ。
Compound angle formulas: sin(A ± B) = sin A cos B ± cos A sin B, cos(A ± B) = cos A cos B ∓ sin A sin B, tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B).
两角和差公式：sin(A ± B) = sin A cos B ± cos A sin B，cos(A ± B) = cos A cos B ∓ sin A sin B，tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)。
Double angle: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ, tan 2θ = 2 tan θ/(1 − tan²θ).
二倍角公式：sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ，tan 2θ = 2 tan θ/(1 − tan²θ)。
Expressions for a cos θ + b sin θ can be written in the form R cos(θ − α) or R sin(θ + α), where R = √(a² + b²), α = arctan(b/a) (with appropriate quadrant).
形如 a cos θ + b sin θ 的表达式可化为 R cos(θ − α) 或 R sin(θ + α)，其中 R = √(a² + b²)，α = arctan(b/a)（需考虑象限）。
Small angle approximations (θ in radians): sin θ ≈ θ, cos θ ≈ 1 − θ²/2, tan θ ≈ θ. These arise from Maclaurin series and are used for small‑angle analysis.
小角近似（θ 以弧度为单位）：sin θ ≈ θ，cos θ ≈ 1 − θ²/2，tan θ ≈ θ。这些近似来自麦克劳林级数，常用于小角度分析。
8. Hyperbolic Functions | 双曲函数
Definitions: sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x = (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ).
定义：sinh x = (eˣ − e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x = (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ)。
Key identity: cosh²x − sinh²x ≡ 1. Also, 1 − tanh²x ≡ sech²x, coth²x − 1 ≡ csch²x.
核心恒等式：cosh²x − sinh²x ≡ 1。此外，1 − tanh²x ≡ sech²x，coth²x − 1 ≡ csch²x。
Osborn’s rule: to convert a trigonometric identity into a hyperbolic identity, replace every product (or implied product) of two sines by the negative of the corresponding hyperbolic product. However, knowing the fundamental definitions and the analogy with sine/cosine often suffices in problem solving.
奥斯本规则：将三角恒等式转换为双曲恒等式时，遇到两个正弦的乘积（或隐含乘积）要添加负号。但在问题中灵活运用定义和与三角函数的类比通常更直接。
Inverse hyperbolic functions: arsinh x = ln(x + √(x²+1)), arcosh x = ln(x + √(x²−1)) for x ≥ 1, artanh x = ½ ln((1+x)/(1−x)) for |x| < 1.
反双曲函数：arsinh x = ln(x + √(x²+1))，arcosh x = ln(x + √(x²−1))（x ≥ 1），artanh x = ½ ln((1+x)/(1−x))（|x| < 1）。
Differentiation: d/dx sinh x = cosh x, d/dx cosh x = sinh x, d/dx tanh x = sech²x.
求导公式：d/dx sinh x = cosh x，d/dx cosh x = sinh x，d/dx tanh x = sech²x。
9. Differentiation and Integration | 微分与积分
Chain rule: dy/dx = dy/du · du/dx. Product rule: d/dx (uv) = u’v + uv’. Quotient rule: d/dx (u/v) = (u’v − uv’)/v².
链式法则：dy/dx = dy/du · du/dx。乘法法则：d/dx (uv) = u’v + uv’。除法法则：d/dx (u/v) = (u’v − uv’)/v²。
Standard derivatives: eˣ, ln x, sin x, cos x, tan x, and reciprocals. Also d/dx (aˣ) = aˣ ln a, d/dx (logₐ x) = 1/(x ln a).
标准函数的导数：eˣ, ln x, sin x, cos x, tan x 及其倒数。此外 d/dx (aˣ) = aˣ ln a, d/dx (logₐ x) = 1/(x ln a)。
Integration as the reverse of differentiation. Standard forms: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫ 1/x dx = ln|x| + C, ∫ eˣ dx = eˣ + C, ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C.
积分是微分的逆运算。常见积分公式：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C（n ≠ −1），∫ 1/x dx = ln|x| + C，∫ eˣ dx = eˣ + C，∫ sin x dx = −cos x + C，∫ cos x dx = sin x + C。
Integration by substitution and integration by parts (∫ u dv = uv − ∫ v du) are essential techniques.
换元积分法和分部积分法（∫ u dv = uv − ∫ v du）是必备积分技巧。
Integrals resulting in inverse trig or hyperbolic functions, e.g. ∫ 1/√(a²−x²) dx = arcsin(x/a) + C, ∫ 1/(a²+x²) dx = (1/a) arctan(x/a) + C.
可化为反三角函数或反双曲函数的积分，例如 ∫ 1/√(a²−x²) dx = arcsin(x/a) + C，∫ 1/(a²+x²) dx = (1/a) arctan(x/a) + C。
Numerical methods: the trapezium rule approximates ∫ₐᵇ f(x) dx ≈ h/2 [y₀ + 2(y₁+…+y_{n-1}) + yₙ] where h = (b−a)/n.
数值积分：梯形法则近似 ∫ₐᵇ f(x) dx ≈ h/2 [y₀ + 2(y₁+…+y_{n-1}) + yₙ]，其中 h = (b−a)/n。
10. Maclaurin Series | 麦克劳林级数
The Maclaurin series expansion of a function f(x) about x = 0 is f(x) = f(0) + f'(0) x + f”(0)/2! x² + f”'(0)/3! x³ + … . It provides a polynomial approximation.
函数 f(x) 在 x = 0 处的麦克劳林级数展开为 f(x) = f(0) + f'(0) x + f”(0)/2! x² + f”'(0)/3! x³ + …，给出了多项式近似。
Common expansions: eˣ = 1 + x + x²/2! + x³/3! + … (all x), sin x = x − x³/3! + x⁵/5! − … (all x), cos x = 1 − x²/2! + x⁴/4! − … (all x), ln(1+x) = x − x²/2 + x³/3 − … (|x| < 1).
常见级数：eˣ = 1 + x + x²/2! + x³/3! + …（对所有 x 成立），sin x = x − x³/3! + x⁵/5! − …（对所有 x），cos x = 1 − x²/2! + x⁴/4! − …（对所有 x），ln(1+x) = x − x²/2 + x³/3 − …（|x| < 1）。
For composite functions, substitute into the standard series, taking care of convergence ranges.
对于复合函数，可代入标准级数展开，同时注意收敛域。
Maclaurin series are used for small‑angle approximations, evaluating limits, and approximating definite integrals.
麦克劳林级数常用于小角近似、求极限以及近似计算定积分。
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