Euler's Identity | 欧拉恒等式

📚 Euler's Identity | 欧拉恒等式

Euler's identity, e^iπ ful 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 e^iπ + 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,

e^iθ = cos θ + i sin θ

real values of θ, and it becomes especially striking when θ = π. Substituting π into Euler's formula gives e^iπ = -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

e^iθ = 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 e^iπ = -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:

e^x = 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! + …

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：

e^x = 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。

广告 / Sponsored

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:

e^iθ = 1 + iθ + (iθ)²/2! + (iθ)³/3! + (iθ)⁴/4! + (iθ)⁵/5! + …

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：

e^iθ = 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

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

广告 / Sponsored

ALevel国际高中数学课程 剑桥牛津培生 CAIE Edexcel AQA国际考试局 课本练习册复习指南 英文教材进…

数学 备考教辅 · 正版现货 · 与本文内容相关

查看详情 / View on Taobao参考价 ¥69.8

更多咨询请联系16621398022（同微信）

备考资讯每日更新，关注不错过

考纲变化 · 科目分析 · 学习方法

在微信里长按二维码识别关注
或截图后用「扫一扫 → 相册」选择本图

公众号：tutorhao