拉普拉斯变换

\[\left\{ \begin{aligned} \lim\limits_{\omega_0\to0}\sum\limits_{-\infty}^{\infty}g(\omega)\frac{1}{T}&=\lim\limits_{\omega_0\to0}\sum\limits_{-\infty}^{\infty}g(\omega)\frac{\omega_0}{2\pi}=\frac{1}{2\pi}\int_{-\infty}^{\infty}g(\omega){\rm d}\omega\\ n\omega_0&\to\omega\\ \lim\limits_{\omega_0\to0}\int_{-\frac{T}{2}}^{\frac{T}{2}}h(t){\rm d}t&=\int_{-\infty}^{\infty}h(t){\rm d}t \end{aligned} \right.\\ \begin{aligned} \Rightarrow f(t)&=\lim\limits_{T\rightarrow\infty}\sum\limits_{n=-\infty}^{\infty}[\frac{1}{T} \int_{0}^{T} f(x)e^{-jn\omega_0 x}{\rm d}x]e^{jn\omega_0 t}\\ &=\frac{1}{2\pi}\int_{-\infty}^{\infty}[\int_{-\infty}^{\infty}f(t)e^{j\omega t}{\rm d}t]e^{j\omega t}{\rm d}\omega\\ &=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(j\omega)e^{j\omega t}{\rm d}\omega \end{aligned} \]

\[F(j\omega) = \int_{-\infty}^{\infty}f(t)e^{-j\omega t}{\rm d}t \]