Exponential Form of a Complex Number | 复数的指数形式

📚 Exponential Form of a Complex Number | 复数的指数形式

The exponential form of a complex number is one of the most elegant and powerful representations in A-Level mathematics. By combining Euler’s formula with the geometric interpretation of complex numbers, it transforms complicated multiplication, division and exponentiation problems into simple manipulations of moduli and arguments. This revision guide covers everything you need for the AQA specification, from definitions to exam-style examples.

复数的指数形式是 A-Level 数学中最优雅、最强大的表示方法之一。它将欧拉公式与复数的几何解释相结合，将复杂的乘法、除法和幂运算转化为对模和辐角的简单操作。本复习指南涵盖 AQA 考纲所要求的全部内容，从定义到考试风格例题。

1. Euler’s Formula | 欧拉公式

Euler’s formula is the foundation of the exponential form. It states that for any real number θ:

欧拉公式是指数形式的基础。它指出，对于任意实数 θ：

e^(iθ) = cos θ + i sin θ

This identity connects exponential functions with trigonometric functions. Substituting θ = π gives Euler’s famous identity e^(iπ) = -1, which beautifully links five fundamental constants: e, i, π, 1 and 0.

这一恒等式将指数函数与三角函数联系起来。代入 θ = π 得到欧拉著名恒等式 e^(iπ) = -1，精妙地将五个基本常数 e、i、π、1 和 0 联系在一起。

Two other useful special cases are e^(iπ/2) = i and e^(-iπ/2) = -i. These arise directly from evaluating cos θ and sin θ at the corresponding angles.

另外两个有用的特殊情况是 e^(iπ/2) = i 与 e^(-iπ/2) = -i。它们可直接通过代入相应角度求 cos θ 和 sin θ 得到。

2. Definition of the Exponential Form | 指数形式的定义

Any complex number z = a + bi, where a and b are real, can be written in exponential form as:

任何复数 z = a + bi（其中 a 和 b 为实数）都可以写成指数形式：

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