Year 13 CCEA Further Mathematics: Formula and Theorem Quick Reference Handbook | Year 13 CCEA 进阶数学：公式定理速查手册

📚 Year 13 CCEA Further Mathematics: Formula and Theorem Quick Reference Handbook | Year 13 CCEA 进阶数学：公式定理速查手册

This quick reference guide brings together all the key formulas, identities and theorems you need for the CCEA Year 13 Further Mathematics course. Covering Further Pure Mathematics topics and essential applied content, it is designed as a one-stop revision sheet to help you memorise and apply the most important results efficiently.

本速查手册汇集了CCEA Year 13进阶数学课程中所有关键的公式、恒等式与定理。内容覆盖进一步纯数和必要的应用模块，旨在通过一张浓缩的复习页帮你高效记忆并运用最重要的结论。

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1. Complex Numbers – Basic Concepts and Formulas | 复数基本概念与公式

A complex number is written as z = x + iy, where x, y are real and i² = –1. The real part is Re(z) = x, the imaginary part is Im(z) = y.

复数记作 z = x + iy，其中 x, y 为实数且 i² = –1。实部 Re(z) = x，虚部 Im(z) = y。

The complex conjugate of z = x + iy is denoted z* = x – iy. Note that z z* = x² + y², which is always real and non‑negative.

复数共轭记为 z* = x – iy。注意 z z* = x² + y² 恒为实数且非负。

The modulus (absolute value) is |z| = √x² + y². Geometrically, |z| is the distance from the origin to the point (x, y) on the Argand diagram.

模（绝对值）为 |z| = √x² + y²。几何上，|z| 表示 Argand 平面上原点到点 (x, y) 的距离。

ies tan θ = y/x, taking care of the quadrant. The principal argument is usually taken in (–π, π] or [0, 2π).

复数 z 的辐角 arg z = θ 满足 tan θ = y/x，需注意象限。主辐角通常取 (–π, π] 或 [0, 2π)。

Properties: |z*| = |z|, arg(z*) = –arg z (mod 2π). Also |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg z₁ + arg z₂.

性质：|z*| = |z|，arg(z*) = –arg z（模 2π）。同时有 |z₁z₂| = |z₁||z₂| 且 arg(z₁z₂) = arg z₁ + arg z₂。

Loci in the complex plane are described by equations. For example, |z – a| = r represents a circle with centre a and radius r. The set given by |z – a| = |z – b| is the perpendicular bisector of the segment joining a and b. The half‑line arg(z – a) = θ is a ray from a making angle θ with the positive real axis.

复平面上的轨迹用方程描述。例如，|z – a| = r 表示以 a 为圆心、r 为半径的圆。方程 |z – a| = |z – b| 表示连接 a 与 b 的线段的垂直平分线。arg(z – a) = θ 代表从 a 出发、与正实轴成 θ 角的射线。

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2. De Moivre's Theorem and Roots of Unity | 棣莫弗定理与单位根

Euler's formula links the exponential form: e^iθ = cos θ + i sin θ. This gives the powerful compact notation z = r e^iθ.

欧拉公式给出了指数形式：e^iθ = cos θ + i sin θ。由此得到紧凑记法 z = r e^iθ。

De Moivre's theorem states that for any rational number n, (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. Using the exponential form, this is simply (e^iθ)ⁿ = e^{i nθ}.

棣莫弗定理指出，对任意有理数 n，有 (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。使用指数形式即 (e^iθ)ⁿ = e^{i nθ}。

To express sin nθ and cos nθ as polynomials in sin θ and cos θ, expand (cos θ + i sin θ)ⁿ via the binomial theorem and equate real and imaginary parts.

要将 sin nθ 和 cos nθ 表达为 sin θ 与 cos θ 的多项式，可将 (cos θ + i sin θ)ⁿ 二项式展开，再分别取实部和虚部。

The n‑th roots of unity are the solutions of zⁿ = 1. They are given by z = e^2πik/n for k = 0, 1, 2, …, n–1. The n roots lie evenly spaced on the unit circle, and their sum is zero.

n 次单位根是方程 zⁿ = 1 的解，由 z = e^2πik/n（k = 0, 1, 2, …, n–1）给出。这 n 个根均匀分布在单位圆上，且它们的和为零。

If a polynomial has real coefficients, any non‑real complex roots occur in conjugate pairs. This helps in factorising polynomials completely over the complex numbers.

若多项式具有实系数，则非实复根必以共轭对出现。这有助于在复数范围内完全分解多项式。

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3. Summation of Finite Series | 有限级数求和公式

The standard results for the sums of powers of integers from 1 to n must be memorised:

必须熟记以下整数幂的和的标准结果：

∑_r=1ⁿ r = n(n+1)/2

∑_r=1ⁿ r² = n(n+1)(2n+1)/6

∑_r=1ⁿ r³ = [n(n+1)/2]²

中文对照：∑_r=1ⁿ r = n(n+1)/2，∑ r² = n(n+1)(2n+1)/6，∑ r³ = [n(n+1)/2]²。

be combined with linearity to sum more complicated polynomials. For instance, ∑(2r³ + 3r² – 5) = 2∑r³ + 3∑r² – 5∑1.

这些公式可与线性性质结合，用于求更复杂的多项式之和。例如，∑(2r³ + 3r² – 5) = 2∑r³ + 3∑r² – 5∑1。

For series expressed in terms of r and n, always break them into separate sums that match the standard forms. Simplifying factorised forms can reveal factorisations like n(n+1)(n+2).

对于用 r 和 n 表示的级数，总是将其拆分为分别匹配标准形式的各项之和。化简后的因式分解常可写成 n(n+1)(n+2) 等形式。

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4. Maclaurin Series Expansions | 麦克劳林级数展开

The Maclaurin series of a function f(x) is given by: f(x) = f(0) + f'(0)x + f''(0)/(2!) x² + f'''(0)/(3!) x³ + …

函数 f(x) 的麦克劳林级数为：f(x) = f(0) + f'(0)x + f''(0)/(2!) x² + f'''(0)/(3!) x³ + …

Standard expansions to learn:

需要学会的标准展开式：

e^x = 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 + … , valid for –1 < x ≤ 1

(1+x)ⁿ = 1 + n x + n(n–1)/(2!) x² + … , valid for |x| < 1 (and any real n)

中文：e^x = 1 + x + x²/2! + …（所有 x）；sin x = x – x³/3! + …（所有 x）；cos x = 1 – x²/2! + …（所有 x）；ln(1+x) = x – x²/2 + x³/3 – …（–1 < x ≤ 1）；(1+x)ⁿ = 1 + nx + …（|x| < 1，n 为任意实数）。

You can derive new series by substitution, differentiation or multiplication. For example, to obtain the series for x sin(2x), substitute 2x into the sine series and multiply by x. The composite argument expansions are used repeatedly in further integration and limits.

通过代换、微分或乘法可推导新级数。例如，求 x sin(2x) 的级数时，将 2x 代入正弦级数并乘以 x。这些复合变元的展开在进一步积分和极限中反复使用。

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5. Hyperbolic Functions and Their Inverses | 双曲函数及其反函数

The hyperbolic functions are defined in terms of exponentials:

双曲函数通过指数定义：

sinh x = (e^x – e^–x)/2,   cosh x = (e^x + e^–x)/2,   tanh x = sinh x / cosh x

双曲正弦 sinh x = (e^x – e^–x)/2，双曲余弦 cosh x = (e^x + e^–x)/2，双曲正切 tanh x = sinh x / cosh x。

The fundamental identity is cosh²x – sinh²x = 1, analogous to the trigonometric identity but with a sign change.

基本恒等式为 cosh²x – sinh²x = 1，与三角恒等式类似但符号不同。

dx(sinh x) = cosh x, d/dx(cosh x) = sinh x, d/dx(tanh x) = sech²x where sech x = 1/cosh x.

导数：d/dx(sinh x) = cosh x，d/dx(cosh x) = sinh x，d/dx(tanh x) = sech²x（其中 sech x = 1/cosh x）。

The inverse hyperbolic functions have logarithmic forms:

反双曲函数可用对数表达：

arsinh x = ln(x + √x² + 1)

arcosh x = ln(x + √x² – 1),   x ≥ 1

artanh x = ½ ln((1+x)/(1–x)),   |x| < 1

c functions mirror the trigonometric ones but without alternating signs: sinh x = x + x³/3! + x⁵/5! + … and cosh x = 1 + x²/2! + x⁴/4! + …

双曲函数的麦克劳林级数与三角函数类似，只是没有正负交替：sinh x = x + x³/3! + x⁵/5! + …；cosh x = 1 + x²/2! + x⁴/4! + …

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6. Relationships Between Roots and Coefficients of Polynomials | 多项式根与系数关系

For a quadratic ax² + bx + c = 0 with roots α and β, we have α + β = –b/a and αβ = c/a.

二次方程 ax² + bx + c = 0 的两根为 α, β，则 α + β = –b/a，αβ = c/a。

For a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ:

三次方程 ax³ + bx² + cx + d = 0 的三根为 α, β, γ:

Σα = α + β + γ = –b/a

Σαβ = αβ + βγ + γα = c/a

αβγ = –d/a

For a quartic ax⁴ + bx³ + cx² + dx + e = 0 with roots α, β, γ, δ:

四次方程 ax⁴ + bx³ + cx² + dx + e = 0 的四根为 α, β, γ, δ:

Σα = –b/a,   Σαβ = c/a,   Σαβγ = –d/a,   αβγδ = e/a

These symmetric sums allow you to find expressions like α²+β²+γ² = (Σα)² – 2Σαβ without solving the equation. Similarly, you can

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