📚 AS CAIE Further Maths: Formulas & Theorems Quick Reference | AS CAIE 进阶数学：公式定理速查手册
This quick reference guide consolidates the essential formulas and theorems for the AS Level CAIE Further Mathematics (9231) syllabus. It covers the core Further Pure Mathematics 1 (FP1) content and highlights key results from the optional applied modules – Further Mechanics and Further Probability & Statistics – so that students can efficiently review all necessary material for their examinations.
本速查手册整理了AS Level CAIE 进阶数学 (9231) 大纲中的基本公式与定理。手册涵盖了核心进阶纯数1 (FP1) 的内容，并突出了应用模块（进阶力学和进阶概率与统计）中的关键结论，以便学生高效复习考试所需的所有材料。
1. Complex Numbers | 复数
Complex numbers extend the real number system by introducing i, where i² = -1. The Cartesian form is written as z = x + iy, with real part Re(z) = x and imaginary part Im(z) = y.
复数通过引入 i（i² = -1）扩展了实数系。笛卡尔形式写作 z = x + iy，实部 Re(z) = x，虚部 Im(z) = y。
z = x + iy
The modulus-argument form expresses a complex number in polar coordinates: z = r(cos θ + i sin θ), where r = |z| = √(x² + y²) and θ = arg z, measured anticlockwise from the positive real axis.
模-辐角形式用极坐标表示复数：z = r(cos θ + i sin θ)，其中 r = |z| = √(x² + y²)，θ = arg z，从正实轴逆时针测量。
z = r(cos θ + i sin θ) = r eⁱθ
Multiplication and division become straightforward in polar form: for z₁ = r₁ eⁱθ₁ and z₂ = r₂ eⁱθ₂, we have z₁z₂ = r₁r₂ eⁱ(θ₁+θ₂) and z₁/z₂ = (r₁/r₂) eⁱ(θ₁–θ₂).
在极坐标形式下乘法和除法变得简单：对于 z₁ = r₁ eⁱθ₁ 和 z₂ = r₂ eⁱθ₂，有 z₁z₂ = r₁r₂ eⁱ(θ₁+θ₂)，z₁/z₂ = (r₁/r₂) eⁱ(θ₁–θ₂)。
De Moivre’s theorem is a powerful tool for finding powers and roots of complex numbers.
棣莫弗定理是求复数的幂和根的有力工具。
(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
To find the n distinct nth roots of a complex number w ≠ 0, first express w in polar form w = R eⁱφ. Then the nth roots are given by zₖ = R^(1/n) eⁱ(φ + 2kπ)/n, for k = 0, 1, …, n−1. These roots lie on a circle of radius R^(1/n) and are equally spaced by angle 2π/n.
要求非零复数 w 的 n 个不同的 n 次方根，先把 w 写成极坐标形式 w = R eⁱφ，则 n 次方根为 zₖ = R^(1/n) eⁱ(φ + 2kπ)/n，k = 0,1,…,n−1。这些根位于半径为 R^(1/n) 的圆上，角度间隔为 2π/n。
2. Roots of Polynomial Equations | 多项式方程的根
For a polynomial equation aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀ = 0, Vieta’s formulas relate the coefficients to symmetric sums of the roots. For a cubic equation ax³ + bx² + cx + d = 0 with roots α, β, γ:
对于多项式方程 aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀ = 0，韦达定理将系数与根的对称和联系起来。对于三次方程 ax³ + bx² + cx + d = 0，根为 α，β，γ：
∑α = α + β + γ = –b/a
∑αβ = αβ + βγ + γα = c/a
αβγ = –d/a
For a quartic equation ax⁴ + bx³ + cx² + dx + e = 0 with roots α, β, γ, δ:
对于四次方程 ax⁴ + bx³ + cx² + dx + e = 0，根为 α，β，γ，δ：
∑α = –b/a
∑αβ = c/a
∑αβγ = –d/a
αβγδ = e/a
We often need to find a new equation whose roots are a simple transformation of the original roots, e.g., kα, α², 1/α, or α + k. The method of substitution replaces x with the appropriate expression and simplifies to obtain the new polynomial.
我们经常需要求一个新方程，其根是原根的简单变换，例如 kα，α²，1/α 或 α + k。代换法用适当的表达式替换 x 并化简以得到新多项式。
3. Matrices and Linear Transformations | 矩阵与线性变换
A 2×2 matrix M = [a b; c d] has determinant det(M) = ad – bc. The matrix is singular if det(M) = 0, meaning it has no inverse.
一个 2×2 矩阵 M = [a b; c d] 的行列式为 det(M) = ad – bc。若 det(M) = 0，矩阵是奇异的，意味着它没有逆矩阵。
M⁻¹ = 1/(ad – bc) · [d –b; –c a]
Eigenvalues λ satisfy the characteristic equation det(M – λI) = 0. For 2×2 matrices, the eigenvectors are non-zero vectors v such that Mv = λv. Invariant lines are lines that map onto themselves, satisfying Mx = λx for some scalar λ, or y = mx + c mapping to itself.
特征值 λ 满足特征方程 det(M – λI) = 0。对于 2×2 矩阵，特征向量是非零向量 v 使得 Mv = λv。不变直线是映射到自身的直线，满足 Mx = λx（λ 为标量），或直线 y = mx + c 映射到自身。
Common transformation matrices include: rotation by angle θ anticlockwise:
常见的变换矩阵包括：逆时针旋转 θ 角：
[cos θ –sin θ; sin θ cos θ]
Reflection in the line y = (tan θ)x:
关于直线 y = (tan θ)x 的反射：
[cos 2θ sin 2θ; sin 2θ –cos 2θ]
4. Mathematical Induction | 数学归纳法
Proof by mathematical induction has three steps: base case (verify the statement for n = 1 or the smallest relevant integer), induction hypothesis (assume true for n = k), and induction step (prove the statement for n = k+1 using the assumption). The conclusion states that the statement holds for all natural numbers n.
数学归纳法证明有三个步骤：基础情形（验证 n = 1 或最小相关整数成立）、归纳假设（假设 n = k 时成立），以及归纳步骤（利用假设证明 n = k+1 成立）。结论是命题对所有自然数 n 成立。
Typical FP1 induction types include summation of series, divisibility, matrix powers, and recurrence relations. For divisibility, you often rewrite f(k+1) as an expression involving f(k) plus a multiple of the divisor.
FP1 中常见的归纳类型包括级数求和、整除性、矩阵幂以及递推关系。对于整除性，通常将 f(k+1) 改写为包含 f(k) 的表达式加上除数的倍数。
5. Summation of Series | 级数求和
Standard sums of the first n natural numbers and their powers are fundamental:
前 n 个自然数及其幂的标准和是基础：
∑ᵣ₌₁ⁿ r = ½ n(n + 1)
∑ᵣ₌₁ⁿ r² = ⅙ n(n + 1)(2n + 1)
∑ᵣ₌₁ⁿ r³ = ¼ n²(n + 1)²
To sum a series of the form ∑ (ar + b)³ or ∑ r(r+1), expand and apply the standard results. The method of differences simplifies series where terms can be expressed as f(r) – f(r+1) (or similar), leading to massive cancellation.
要求如 ∑ (ar + b)³ 或 ∑ r(r+1) 这样的级数和，可展开并应用标准结果。差分法在项可表示为 f(r) – f(r+1)（或类似）时化简级数，带来大量消元。
∑ᵣ₌₁ⁿ [f(r) – f(r+1)] = f(1) – f(n+1)
6. Maclaurin Series | 麦克劳林级数
The Maclaurin series expansion of a function f(x) about x = 0 is given by:
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)x²/2! + … + f⁽ⁿ⁾(0)xⁿ/n! + …
Standard Maclaurin series and their intervals of validity are essential to know.
必须熟记标准麦克劳林级数及其收敛区间。
eˣ = 1 + x + x²/2! + x³/3! + … , valid for all x
sin x = x – x³/3! + x⁵/5! – … , valid for all x
cos x = 1 – x²/2! + x⁴/4! – … , valid for all x
ln(1 + x) = x – x²/2 + x³/3 – x⁴/4 + … , –1 < x ≤ 1
(1 + x)ⁿ = 1 + nx + n(n–1)x²/2! + … , |x| < 1 (n rational)
Composite functions can be expanded by substituting into these series, provided the final x-range is adjusted to ensure validity.
复合函数可以通过代入这些级数来展开，但需调整最终的 x 范围以确保收敛。
7. Hyperbolic Functions | 双曲函数
Hyperbolic functions are defined in terms of exponentials:
双曲函数用指数函数定义：
sinh x = (eˣ – e⁻ˣ)/2
cosh x = (eˣ + e⁻ˣ)/2
tanh x = sinh x / cosh x
Key identities mirror trigonometric ones, often with sign differences. Osborne’s rule helps convert trigonometric identities to hyperbolic identities: replace each cos with cosh, sin with i sinh, and flip the sign of any term involving a product of two sines.
关键恒等式与三角恒等式相似，但常伴随符号差异。奥斯本法则有助于将三角恒等式转换为双曲恒等式：将 cos 替换为 cosh，sin 替换为 i sinh，并翻转任何包含两个正弦乘积项的符号。
cosh² x – sinh² x = 1
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh² x + sinh² x = 2 cosh² x – 1 = 1 + 2 sinh² x
The inverse hyperbolic functions can be expressed as natural logarithms:
反双曲函数可表示为自然对数：
arsinh x = ln(x + √(x² + 1)), for all x
arcosh x = ln(x + √(x² – 1)), x ≥ 1
artanh x = ½ ln((1 + x)/(1 – x)), |x| < 1
8. Further Mechanics – Linear Kinematics & Momentum | 进阶力学 – 线运动学与动量
For constant acceleration in one dimension, the SUVAT equations apply:
对于一维匀加速运动，适用 SUVAT 方程：
v = u + at
s = ut + ½ at²
v² = u² + 2as
s = ½ (u + v)t
When acceleration varies, use calculus: displacement s(t), velocity v = ds/dt, acceleration a = dv/dt = d²s/dt². For vector motion in 2D, treat the i- and j-components separately using the same kinematic relations.
当加速度变化时，使用微积分：位移 s(t)，速度 v = ds/dt，加速度 a = dv/dt = d²s/dt²。对于二维矢量运动，分别处理 i 和 j 分量，使用同样的运动学关系。
Momentum is defined as p = mv. Impulse I equals the change in momentum: I = mv – mu. The principle of conservation of linear momentum states that the total momentum of a system remains constant if no external resultant force acts.
动量定义为 p = mv。冲量 I 等于动量的变化：I = mv – mu。线动量守恒原理指出，若系统不受合外力作用，总动量保持不变。
For direct collisions between two particles, Newton’s law of restitution relates relative speeds:
对于两质点的对心碰撞，牛顿恢复系数定律关联相对速度：
e = (v₂ – v₁) / (u₁ – u₂)
where e is the coefficient of restitution (0 ≤ e ≤ 1), and u, v denote pre- and post-impact velocities. For oblique collisions against a smooth fixed surface, the velocity component parallel to the surface is unchanged, while the perpendicular component is reversed and multiplied by e.
其中 e 是恢复系数 (0 ≤ e ≤ 1)，u 和 v 分别表示碰撞前和后的速度。对于与光滑固定面的斜碰，平行于面的速度分量不变，垂直于面的分量反向并乘以 e。
9. Further Mechanics – Work, Energy & Power | 进阶力学 – 功、能与功率
Work done by a constant force F moving its point of application through displacement s in the direction of the force is W = Fs. When the force varies with position, use integration.
恒力 F 使其作用点沿力的方向发生位移 s，所做的功为 W = Fs。当力随位置变化时，使用积分。
W = ∫ F dx
Kinetic energy (KE) = ½ mv², and gravitational potential energy (GPE) = mgh where h is the height above a chosen zero level. The work-energy principle states that the total work done by all forces (external and resistive) equals the change in kinetic energy.
动能 (KE) = ½ mv²，重力势能 (GPE) = mgh，h 为相对于选定零势能面的高度。功能原理指出，所有力（外力和阻力）做的总功等于动能的变化量。
W_total = ΔKE
If only conservative forces
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