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歐拉恆等式 e^(iπ) + 1 = 0：泰勒級數如何將指數與三角波統一？

When the compound-growth constant e meets the imaginary unit i, runaway exponential growth transforms into eternal rotation on the unit circle.

e^{i\theta} = \cos\theta + i\sin\theta \implies e^{i\pi} + 1 = 0

Multiplying by i rotates a complex vector by 90°, turning exponential growth into constant-speed circular motion.

Maclaurin series expansion of e^(iθ) naturally separates into even real powers (cos θ) and odd imaginary powers (i sin θ).

Replaces cumbersome trigonometric identities with simple exponent addition, powering Fourier analysis and quantum mechanics.

⚡ 本文涉及的核心數學公式

#C01極限的定義與自然常數 e 的誕生 #G04單位圓三角函數與基本恆等式 #A09虛數單位 i 與複平面極座標形式 #C06泰勒級數與麥克勞林多項式展開 #C07歐拉公式與歐拉恆等式 (上帝公式)#C09傅立葉級數與傅立葉變換 (時域↔頻域)

1. Alternating Signs in the Maclaurin Expansion

e^{i\theta} = \sum_{n=0}^{\infty} \frac{(i\theta)^n}{n!} = \left(1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \cdots\right) + i\left(\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \cdots\right)

Substituting x = iθ into the Maclaurin series of e^x introduces the 4-cycle of powers of i (1, i, -1, -i), cleanly splitting the infinite sum into the real series for cos θ and the imaginary series for sin θ.

2. Differential Intuition: Why Rotation Instead of Growth?

\frac{d}{d\theta} e^{i\theta} = i e^{i\theta} \implies \text{Velocity } \perp \text{ Position}

In the complex plane, multiplication by i rotates a vector by 90° counterclockwise. Because the derivative d/dθ(e^(iθ)) = i·e^(iθ) is always perpendicular to the position vector, the point traces a unit circle of radius 1.

3. Gateway to Fourier Transform and Modern DSP

\hat{f}(\omega) = \int_{-\infty}^{\infty} f(t)\,e^{-i\omega t}\,dt

Euler's formula packs both amplitude and phase into a single rotating complex exponential e^(-iωt), enabling Fast Fourier Transform (FFT), Wi-Fi OFDM modulation, and MRI k-space reconstruction.