📚 Euler’s Identity | 欧拉恒等式
Euler’s identity, eiπ + 1 = 0, is often called the most beautiful equation in mathematics. It brings together the natural exponential base e, the imaginary unit i, the circle constant π, the number 1, and the number 0 in one concise statement. In this article, we will derive the identity from Euler’s formula and from the series expansions encountered in A-Level Mathematics.
欧拉恒等式 eiπ + 1 = 0 常被誉为数学中最优美的方程。它将自然对数的底 e、虚数单位 i、圆周率 π、数字 1 与数字 0 统一在一个简洁的式子中。本文将利用欧拉公式以及 A-Level 数学中学过的级数展开推导这个恒等式。
1. Euler’s Formula | 欧拉公式
Euler’s formula is the general result that for any real angle θ, measured in radians,
eiθ = cos θ + i sin θ
This formula creates a bridge between exponential functions and trigonometric functions. It is valid for all real values of θ, and it becomes especially striking when θ = π. Substituting π into Euler’s formula gives eiπ = -1, which immediately rearranges to the famous identity.
欧拉公式是一个一般性结论：对任意以弧度为单位的角度 θ，有
eiθ = cos θ + i sin θ
这一公式在指数函数和三角函数之间架起了桥梁。它对所有实数 θ 都成立，而当 θ = π 时尤其引人注目。将 π 代入欧拉公式得到 eiπ = -1，稍加整理便得到这个著名的恒等式。
2. Maclaurin Expansions | 麦克劳林展开
The Maclaurin series for the exponential and trigonometric functions are:
ex = 1 + x + x²/2! + x³/3! + x⁴/4! + …
sin x = x – x³/3! + x⁵/5! – x⁷/7! + …
cos x = 1 – x²/2! + x⁴/4! – x⁶/6! + …
These series can be obtained by repeatedly differentiating the functions and evaluating them at x = 0. In AQA A-Level Mathematics, you are expected to use the exponential series for approximations. The same expansions will now open the door to Euler’s identity.
指数函数和三角函数的麦克劳林级数分别为：
ex = 1 + x + x²/2! + x³/3! + x⁴/4! + …
sin x = x – x³/3! + x⁵/5! – x⁷/7! + …
cos x = 1 – x²/2! + x⁴/4! – x⁶/6! + …
这些级数可以通过反复求导并在 x = 0 处取值来得到。在 AQA A-Level 数学中，你需要会用指数级数进行近似计算。接下来，同样的展开将成为推导欧拉恒等式的钥匙。
3. Substituting x = iθ | 代入 x = iθ
Since the exponential series is absolutely convergent, we may replace the real variable x with the complex number iθ. This yields:
eiθ = 1 + iθ + (iθ)²/2! + (iθ)³/3! + (iθ)⁴/4! + (iθ)⁵/5! + …
To simplify this expression, we use the cyclic powers of i: i² = -1, i³ = -i, i⁴ = 1, i⁵ = i, and so on. The pattern repeats every four powers.
由于指数级数绝对收敛，我们可以将实变量 x 替换为复数 iθ。于是得到：
eiθ = 1 + iθ + (iθ)²/2! + (iθ)³/3! + (iθ)⁴/4! + (iθ)⁵/5! + …
为化简该式，我们利用 i 的循环幂：i² = -1，i³ = -i，i⁴ = 1，i⁵ = i，依此类推。这个规律每四次重复一次。
4. Grouping Real and Imaginary Parts | 分出实部与虚部
Now write out the first few powers:
(iθ)² = i²θ² = -θ²
(iθ)³ = i³θ³ = -iθ³
(iθ)⁴ = i
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