Year 13 Edexcel Further Maths: Quick Reference Handbook of Formulas & Theorems | Year 13 Edexcel 进阶数学：公式定理速查手册

📚 Year 13 Edexcel Further Maths: Quick Reference Handbook of Formulas & Theorems | Year 13 Edexcel 进阶数学：公式定理速查手册

This concise handbook gathers the essential formulas, theorems and standard results from the Year 13 Edexcel Further Mathematics course, covering Core Pure 1 and Core Pure 2. Each section presents key ideas in paired English and Chinese paragraphs so that you can quickly locate and revise the concepts needed for A‑level examinations. All expressions use standard mathematical notation with Unicode symbols and no LaTeX – ready for easy reading and memorisation.

这本速查手册汇集了 Year 13 Edexcel 进阶数学课程（核心纯数 1 与核心纯数 2）中的关键公式、定理和标准结果。每个小节均以配对的英文和中文段落呈现核心内容，帮助您快速定位并复习 A‑level 考试中需要的所有概念。所有表达式采用标准数学符号和 Unicode 字符书写，不含 LaTeX，便于阅读与记忆。

1. Complex Numbers – Polar Form & Exponential Form | 复数 – 极坐标形式与指数形式

A complex number z = x + iy can be written in polar form using the modulus r = |z| = √(x² + y²) and argument θ = arg(z). The exponential form uses Euler’s relation e^iθ = cos θ + i sin θ.

复数 z = x + iy 可用模 r = |z| = √(x² + y²) 和辐角 θ = arg(z) 写成极坐标形式。指数形式则利用欧拉公式 e^iθ = cos θ + i sin θ。

z = r(cos θ + i sin θ) = r e^iθ

Multiplication in polar form: z₁z₂ = r₁r₂ e^{i(θ₁+θ₂)}
Division: z₁/z₂ = (r₁/r₂) e^{i(θ₁−θ₂)}

极坐标形式乘法：z₁z₂ = r₁r₂ e^{i(θ₁+θ₂)}；除法：z₁/z₂ = (r₁/r₂) e^{i(θ₁−θ₂)}。

The argument satisfies θ ∈ (−π, π] for the principal argument. Conjugate: z̄ = r e^−iθ.

主辐角取值满足 θ ∈ (−π, π]。共轭复数：z̄ = r e^−iθ。

2. De Moivre’s Theorem & Trigonometric Applications | 棣莫弗定理与三角应用

De Moivre’s theorem states that for any integer n: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. It is the cornerstone for deriving multiple‑angle formulas and powers of complex numbers.

棣莫弗定理指出，对任意整数 n 有 (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。该定理是推导倍角公式和复数幂的基础。

zⁿ = [r e^iθ]ⁿ = rⁿ e^{i nθ}

From De Moivre, we can express cos nθ and sin nθ as polynomials in cos θ and sin θ. To find all n-th roots of a complex number w = s e^iφ: z_k = s^1/n e^i(φ+2kπ)/n, k = 0, 1, …, n−1.

利用棣莫弗定理，可将 cos nθ 与 sin nθ 表示为 cos θ 和 sin θ 的多项式。求复数 w = s e^iφ 的全部 n 次方根时使用：z_k = s^1/n e^i(φ+2kπ)/n, k = 0, 1, …, n−1。

3. Summation of Series & Method of Differences | 级数求和与差分法

Standard summations are required for the method of differences and other series manipulations. Using ∑_r=1ⁿ r, r², r³, we can tackle more complicated sums.

差分法和其他级数运算需要标准求和公式。利用 ∑_r=1ⁿ r、r²、r³，可以计算更复杂的求和。

∑_r=1ⁿ r = n(n+1)/2
∑_r=1ⁿ r² = n(n+1)(2n+1)/6
∑_r=1ⁿ r³ = n²(n+1)²/4

The method of differences applies when a term u_r can be written as f(r) − f(r−1) or f(r+1)−f(r). Then the sum telescopes: ∑_r=1ⁿ u_r = f(n) − f(0).

若通项 u_r 可写成 f(r) − f(r−1) 或 f(r+1)−f(r)，即可使用差分法。此时求和可裂项相消：∑_r=1ⁿ u_r = f(n) − f(0)。

Often partial fractions are used to create a difference pattern, e.g. 1/[r(r+1)] = 1/r − 1/(r+1).

通常借助部分分式构造差分形式，例如 1/[r(r+1)] = 1/r − 1/(r+1)。

4. Matrices – Eigenvalues and Eigenvectors | 矩阵 – 特征值与特征向量

For a square matrix A, a non‑zero vector v is an eigenvector if A v = λ v for some scalar λ, the eigenvalue. The eigenvalues are found from the characteristic equation det(A − λI) = 0.

对于方阵 A，若存在非零向量 v 满足 A v = λ v，则 v 为特征向量，标量 λ 为特征值。特征值由特征方程 det(A − λI) = 0 求得。

det(A − λI) = 0

Once eigenvalues λ_i are known, corresponding eigenvectors are obtained by solving (A − λ_iI) v = 0. For 2×2 matrices the eigenvalues can be found from λ² − tr(A)λ + det(A) = 0.

求出特征值 λ_i 后，通过求解 (A − λ_iI) v = 0 得到相应的特征向量。对于 2×2 矩阵，特征值可由 λ² − tr(A)λ + det(A) = 0 得出。

If A has n independent eigenvectors, it can be diagonalised: P⁻¹AP = D, where D is diagonal with eigenvalues on the leading diagonal and columns of P are the corresponding eigenvectors.

若 A 有 n 个线性无关的特征向量，则可对角化：P⁻¹AP = D，其中 D 为对角矩阵，主对角线元素为特征值，P 的列是对应的特征向量。

Property	Formula
Inverse of 2×2 matrix	(1/det(A)) [d −b; −c a]

2×2 矩阵的逆：A⁻¹ = (1/det(A)) [d −b; −c a]。

5. Hyperbolic Functions | 双曲函数

Hyperbolic functions are defined in terms of exponentials: sinh x = (e^x − e^−x)/2, cosh x = (e^x + e^−x)/2. The fundamental identity is cosh²x − sinh²x = 1.

双曲函数通过指数函数定义：sinh x = (e^x − e^−x)/2，cosh x = (e^x + e^−x)/2。基本恒等式为 cosh²x − sinh²x = 1。

cosh²x − sinh²x = 1

Their derivatives mirror trigonometric ones but without sign changes: d/dx(sinh x) = cosh x, d/dx(cosh x) = sinh x. The inverse hyperbolic functions are logarithmic: arsinh x = ln(x + √(x²+1)), arcosh x = ln(x + √(x²−1)) for x ≥ 1.

它们的导数与三角函数的类似但无符号变化：d/dx(sinh x) = cosh x，d/dx(cosh x) = sinh x。反双曲函数为对数形式：arsinh x = ln(x + √(x²+1))，arcosh x = ln(x + √(x²−1)) (x ≥ 1)。

Function	Derivative
sinh x	cosh x
cosh x	sinh x
tanh x	sech²x

6. Further Calculus – Standard Integrals & Differentiation | 进阶微积分 – 标准积分与微分

You must recall derivatives of inverse trigonometric and hyperbolic functions, as well as integrals that produce such functions. Often these appear after completing the square or using substitution.

必须牢记反三角函数和反双曲函数的导数，以及能积分得到这些函数的标准积分。它们常出现在配方或换元之后。

d/dx(arcsin x) = 1/√(1 − x²)
d/dx(arctan x) = 1/(1 + x²)

Key integrals:

关键积分：

Integral	Result
∫ 1/√(a² − x²) dx	arcsin(x/a) + C
∫ 1/(a² + x²) dx	(1/a) arctan(x/a) + C
∫ 1/√(x² ± a²) dx	arsinh(x/a) + C or arcosh(x/a) + C (for + sign, x > a)

For rational functions, use partial fractions. Integration by parts follows ∫ u dv = uv − ∫ v du.

对于有理函数，使用部分分式分解。分部积分法：∫ u dv = uv − ∫ v du。

7. First-Order Differential Equations | 一阶微分方程

In Core Pure 2, first‑order equations are mainly linear: dy/dx + P(x) y = Q(x). The integrating factor is I(x) = e^{∫P dx}. Multiply through by I(x) to write d/dx (I y) = I Q, then integrate.

核心纯数 2 中主要为一阶线性微分方程：dy/dx + P(x) y = Q(x)。积分因子为 I(x) = e^{∫P dx}。方程两边乘 I(x) 后可写成 d/dx (I y) = I Q，再积分求解。

y I = ∫ I Q dx + C

Some applied contexts (e.g. cooling, mixing) yield separable equations dy/dx = f(x)g(y), which are solved by rearranging to (1/g(y)) dy = f(x) dx and integrating both sides.

一些应用背景（如冷却、混合）会产生可分离变量方程 dy/dx = f(x)g(y)，通过变形为 (1/g(y)) dy = f(x) dx 并两边积分求解。

8. Second-Order Differential Equations | 二阶微分方程

The general homogeneous linear equation is a d²y/dx² + b dy/dx + c y = 0, with constant coefficients. The auxiliary equation is a m² + b m + c = 0. The form of the complementary function depends on the roots.

一般齐次线性方程为 a d²y/dx² + b dy/dx + c y = 0，系数为常数。辅助方程为 a m² + b m + c = 0。余函数的形式取决于根的类型。

a m² + b m + c = 0

Real distinct roots m₁, m₂: y_c = A e^m₁x + B e^m₂x.
Repeated root m: y_c = (A + B x) e^mx.
Complex roots α ± iβ: y_c = e^αx (A cos βx + B sin βx).

实不等根 m₁, m₂：y_c = A e^m₁x + B e^m₂x；重根 m：y_c = (A + B x) e^mx；复根 α ± iβ：y_c = e^αx (A cos βx + B sin βx)。

For the non‑homogeneous equation a y” + b y’ + c y = f(x), find a particular integral y_p by trial (polynomial, exponential, trigonometric). General solution: y = y_c + y_p.

对于非齐次方程 a y” + b y’ + c y = f(x)，通过试函数法（多项式、指数、三角）求特解 y_p。通解为 y = y_c + y_p。

9. Polar Coordinates – Areas and Curves | 极坐标 – 面积与曲线

A curve can be defined by r = f(θ). The area enclosed by a polar curve between θ = α and θ = β is (1/2) ∫_α^β r² dθ. For a loop, you need to find the limits where r = 0.

曲线可以由 r = f(θ) 定义。极曲线在 θ = α 到 θ = β 之间围成的面积为 (1/2) ∫_α^β r² dθ。求环状区域面积时，需先找到 r = 0 对应的角度作为积分限。

A = ½ ∫_α^β r² dθ

Common shapes include cardioids r = a(1 + cos θ), limacons and roses r = a cos nθ. Tangents parallel or perpendicular to the initial line are found from dy/dx = (r’ sin θ + r cos θ)/(r’ cos θ − r sin θ).

常见曲线有心形线 r = a(1 + cos θ)、蚶线和玫瑰线 r = a cos nθ。切线平行或垂直于极轴时，可利用 dy/dx = (r’ sin θ + r cos θ)/(r’ cos θ − r sin θ) 来判断。

10. Taylor and Maclaurin Series | 泰勒与麦克劳林级数

Maclaurin’s series expands a function about x = 0: f(x) = f(0) + f'(0) x + f”(0)/2! x² + f”'(0)/3! x³ + … Taylor’s series generalises to expansion about x = a: f(x) = f(a) + f'(a)(x − a) + f”(a)/2! (x − a)² + …

麦克劳林级数在 x = 0 处展开：f(x) = f(0) + f'(0) x + f”(0)/2! x² + f”'(0)/3! x³ + …。泰勒级数将其推广到 x = a 处：f(x) = f(a) + f'(a)(x − a) + f”(a)/2! (x − a)² + …。

e^x = 1 + x + x²/2! + x³/3! + …
sin x = x − x³/3! + x⁵/5! − …
cos x = 1 − x²/2! + x⁴/4! − …
ln(1 + x) = x − x²/2 + x³/3 − … (|x| < 1)

You can compose series or differentiate/integrate term‑by‑term within the radius of convergence. These expansions are essential for evaluating limits and approximating functions.

可在收敛半径内进行级数的复合或逐项微积分。这些展开式在求极限与近似计算中至关重要。

11. Numerical Methods – Newton‑Raphson, Euler’s Method | 数值方法 – 牛顿‑拉弗森迭代法与欧拉法

The Newton‑Raphson method iteratively finds roots of f(x) = 0 using x_n+1 = x_n − f(x_n)/f'(x_n). It usually converges rapidly near the root if f'(x) ≠ 0.

牛顿‑拉弗森法通过迭代 x_n+1 = x_n − f(x_n)/f'(x_n) 求解 f(x) = 0 的根。当 f'(x) ≠ 0 且在根附近时通常收敛很快。

x_n+1 = x_n − f(x_n)/f'(x_n)

For first‑order differential equations dy/dx = f(x, y) with initial condition (x₀, y₀), Euler’s method steps forward: y_n+1 = y_n + h f(x_n, y_n), with step size h.

对于一阶微分方程 dy/dx = f(x, y) 及初始条件 (x₀, y₀)，欧拉法进行递推：y_n+1 = y_n + h f(x_n, y_n)，步长为 h。

y_n+1 = y_n + h f(x_n, y_n)

Mid‑point formula (improved Euler) gives better accuracy: y_n+1 = y_n + h f(x_n + h/2, y_n + (h/2) f(x_n, y_n)).

中点公式（改进欧拉法）精度更高：y_n+1 = y_n + h f(x_n + h/2, y_n + (h/2) f(x_n, y_n))。

12. Vectors – Scalar Product, Equations of Lines and Planes | 向量 – 数量积、直线与平面方程

The scalar (dot) product of vectors a and b is a · b = |a||b| cos θ = a₁

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Year 13 Edexcel Further Maths: Quick Reference Handbook of Formulas & Theorems | Year 13 Edexcel 进阶数学：公式定理速查手册