📚 Edexcel Further Maths Year 2 | 爱德思进阶数学第二年
The Edexcel Further Maths Year 2 course builds on the pure mathematics met in Year 1 and introduces the deeper tools needed for advanced problem solving in mathematics, physics and engineering. This revision guide summarises the most common FP2-style topics, including complex numbers, hyperbolic functions, polar coordinates, differential equations, series and matrices.
爱德思进阶数学第二年课程建立在第一年纯数学的基础上，引入更深入的数学工具，用于数学、物理和工程中的高阶问题求解。本复习指南总结了 FP2 最常见的话题，包括复数、双曲函数、极坐标、微分方程、级数和矩阵。
1. Complex Numbers and De Moivre's Theorem | 复数与棣莫弗定理
s-argument form. If z has modulus r and argument θ, then z = r(cos θ + i sin θ). De Moivre's theorem says that (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ for any integer n.
在第二年，你将复数从笛卡尔形式 a + bi 扩展到模-辐角形式。如果复数 z 的模为 r、辐角为 θ，则 z = r(cos θ + i sin θ)。棣莫弗定理指出，对于任意整数 n，有 (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ。
zⁿ = rⁿ (cos nθ + i sin nθ)
arg w + 2kπ)/n for k = 0, 1, ..., n − 1. The roots lie symmetrically on a circle of radius |w|¹ᐟⁿ.
该定理可用于求复数的幂和 n 次方根。解方程 zⁿ = w 时，先将 w 写成模-辐角形式，并在辐角上加上 2π 的整数倍：θ = (arg w + 2kπ)/n，其中 k = 0, 1, ..., n − 1。这些根均匀分布在半径为 |w|¹ᐟⁿ 的圆上。
2. Hyperbolic Functions | 双曲函数
The hyperbolic functions are defined in terms of exponential functions. The three basic ones are sinh x, cosh x and tanh x, which behave like trigonometric functions but with important sign differences.
双曲函数由指数函数定义。三个基本双曲函数是 sinh x、cosh x 和 tanh x，它们与三角函数相似，但有重要的符号差异。
sinh x = (eˣ − e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, tanh x = sinh x / cosh x
as d/dx sinh x = cosh x and d/dx cosh x = sinh x, as well as the inverse derivative forms.
关键恒等式是 cosh²x − sinh²x = 1。你还需要掌握导数，例如 d/dx sinh x = cosh
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