Using de Moivre’s Theorem to Evaluate Powers of Complex Numbers | 使用德莫弗定理计算复数的幂

📚 Using de Moivre’s Theorem to Evaluate Powers of Complex Numbers | 使用德莫弗定理计算复数的幂

De Moivre’s theorem is one of the most powerful tools in the AQA A-Level Further Mathematics syllabus. Named after the French mathematician Abraham de Moivre, it states that for any real number n, (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). This elegant result transforms the evaluation of powers of complex numbers from laborious algebraic expansion into straightforward trigonometric evaluation.

德莫弗定理是 AQA A-Level 进阶数学教学大纲中最强大的工具之一。它以法国数学家亚伯拉罕·德莫弗的名字命名，指出对于任何实数 n，(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)。这个优雅的结果将复数幂的计算从繁重的代数展开转化为直接的三角函数求值。

1. The Polar Form of a Complex Number | 复数的极坐标形式

Before applying de Moivre’s theorem, we must be completely comfortable writing complex numbers in polar form. A complex number z = x + iy can be expressed as z = r(cos θ + i sin θ), where r = |z| = √(x² + y²) is the modulus and θ = arg(z) is the argument measured in radians from the positive real axis.

在运用德莫弗定理之前，我们必须熟练掌握复数的极坐标形式。复数 z = x + iy 可以表示为 z = r(cos θ + i sin θ)，其中 r = |z| = √(x² + y²) 是模长，θ = arg(z) 是从正实轴起按弧度计量的辐角。

For a point in the Argand diagram, the principal argument Arg(z) lies in the interval (-π, π]. When the complex number lies in the second or third quadrant, we must add or subtract π to obtain the correct angle. The shorthand notation z = r cis θ is often used, where cis θ means cos θ + i sin θ.

对于阿尔冈图中的一个点，主辐角 Arg(z) 位于区间 (-π, π] 内。当复数位于第二或第三象限时，我们必须加上或减去 π 才能得到正确角度。常用简写记号 z = r cis θ，其中 cis θ 表示 cos θ + i sin θ。

2. Statement of de Moivre’s Theorem | 德莫弗定理的表述

De Moivre’s theorem states that for any real number n and any angle θ:

德莫弗定理指出，对于任何实数 n 和任何角度 θ：

(cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)

Equivalently, if z = r cis θ, then zⁿ = rⁿ cis(nθ) = rⁿ[cos(nθ) + i sin(nθ)]. The theorem holds for all integer values of n — positive, negative, and zero — and, in its extended form, for all real n.

等价地，若 z = r cis θ，则 zⁿ = rⁿ cis(nθ) = rⁿ[cos(nθ) +

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