欧拉公式:
\[ e^{i\theta}=\cos \theta + i \sin \theta \]
证明一
令
\[ f(\theta)=\frac{e^{i\theta}}{\cos \theta + i \sin \theta} \]
对 \(f(\theta)\) 求导,可以得到:
\[ \begin{aligned} f^{\prime}(\theta) &= \frac{\left(e^{i\theta}\right)^{\prime}(\cos \theta + i \sin \theta)-e^{i\theta}\left(\cos \theta + i \sin \theta\right)^\prime}{\left(\cos \theta + i \sin \theta\right)^2}\\ &= \frac{i\cdot e^{i\theta}(\cos \theta + i \sin \theta)-e^{i\theta}\left(-\sin \theta + i \cos \theta\right)}{\left(\cos \theta + i \sin \theta\right)^2}\\ &= \frac{i\cdot e^{i\theta}(\cos \theta + i \sin \theta)-i\cdot e^{i\theta}\left(\cos \theta +i \sin \theta\right)}{\left(\cos \theta + i \sin \theta\right)^2}\\ &=0 \end{aligned} \]
故 \(f(\theta)\) 为常数,\(f(\theta)=f(0)=1\)。证毕。
证明二
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