\[
f_1(t) * f_2(t) = \int_0^t f_1(t-\tau) f_2(\tau) \mathrm{d}\tau
\]
由于 \(\tau > t\) 时,有 \(f_1(t-\tau)1(t-\tau)=0\),因此
\[
\int_0^t f_1(t-\tau) f_2(\tau) \mathrm{d}\tau = \int_0^\infty f_1(t-\tau)1(t-\tau) f_2(\tau) \mathrm{d}\tau
\]
于是
\[
\mathscr{L}\left[\int_0^t f_1(t-\tau) f_2(\tau) \mathrm{d}\tau\right] = \mathscr{L}\left[\int_0^\infty f_1(t-\tau)1(t-\tau) f_2(\tau) \mathrm{d}\tau\right]
\]
\[
= \int_0^\infty \mathrm{e}^{-st}\left[\int_0^\infty f_1(t-\tau)1(t-\tau) f_2(\tau) \mathrm{d}\tau\right]\mathrm{d}t
\]
令 \(t-\tau=\lambda\),并改变积分次序,可得
\[
\mathscr{L}\left[\int_0^t f_1(t-\tau) f_2(\tau) \mathrm{d}\tau\right] = \int_0^\infty f_1(t-\tau)1(t-\tau)\mathrm{e}^{-st}\mathrm{d}t \int_0^\infty f_2(\tau)\mathrm{d}\tau
\]
\[
= \int_0^\infty f_1(\lambda)\mathrm{e}^{-s(\lambda+\tau)}\mathrm{d}\lambda \int_0^\infty f_2(\tau)\mathrm{d}\tau
\]
\[
= \int_0^\infty f_1(\lambda)\mathrm{e}^{-s\lambda}\mathrm{d}\lambda \int_0^\infty f_2(\tau)\mathrm{e}^{-s\tau}\mathrm{d}\tau
\]
\[
= F_1(s) F_2(s)
\]
式中
\[
F_1(s) = \int_0^\infty f_1(t)\mathrm{e}^{-st}\mathrm{d}t = \mathscr{L}[f_1(t)], \qquad F_2(s) = \int_0^\infty f_2(t)\mathrm{e}^{-st}\mathrm{d}t = \mathscr{L}[f_2(t)]
\]
拉普拉斯变换的基本性质,如表 1-2 所示。
表 1-2 拉普拉斯变换的基本性质
| 序号 | 基本运算 | \(f(t)\) | \(F(s)=\mathscr{L}[f(t)]\) |
|---|---|---|---|
| 1 | 拉普拉斯变换定义 | \(f(t)\) | \(F(s)=\displaystyle\int_0^\infty f(t)\mathrm{e}^{-st}\mathrm{d}t\) |
| 2 | 位移(时间域) | \(f(t-\tau_0)1(t-\tau_0)\) | \(\mathrm{e}^{-\tau_0 s}F(s), \ \tau_0>0\) |
| 3 | 相似性 | \(f(at)\) | \(\dfrac{1}{a}F\left(\dfrac{s}{a}\right), \ a>0\) |
| 4 | 一阶导数 | \(\dfrac{\mathrm{d}f(t)}{\mathrm{d}t}\) | \(sF(s)-f(0)\) |
| 5 | \(n\) 阶导数 | \(\dfrac{\mathrm{d}^n f(t)}{\mathrm{d}t^n}\) | \(s^nF(s)-s^{n-1}f(0)-s^{n-2}\dot f(0)-\cdots-f^{(n-1)}(0)\) |
| 6 | 不定积分 | \(\displaystyle\int f(t)\mathrm{d}t\) | \(\dfrac{1}{s}\left[F(s)+f^{-1}(0)\right]\) |
| 7 | 定积分 | \(\displaystyle\int_0^t f(t)\mathrm{d}t\) | \(\dfrac{1}{s}F(s)\) |
| 8 | 函数乘以 \(t\) | \(tf(t)\) | \(-\dfrac{\mathrm{d}}{\mathrm{d}s}F(s)\) |
| 9 | 函数除以 \(t\) | \(\dfrac{1}{t}f(t)\) | \(\displaystyle\int_t^\infty F(s)\mathrm{d}s\) |
| 10 | 位移(\(s\) 域) | \(\mathrm{e}^{-at}f(t)\) | \(F(s+a)\) |
| 11 | 初始值 | \(\displaystyle\lim_{t\to 0_+} f(t)\) | \(\displaystyle\lim_{s\to\infty} sF(s)\) |
| 12 | 终值 | \(\displaystyle\lim_{t\to\infty} f(t)\) | \(\displaystyle\lim_{s\to 0} sF(s)\) |
| 13 | 卷积 | \(f_1(t)*f_2(t)=\displaystyle\int_0^t f_1(\tau)f_2(t-\tau)\mathrm{d}\tau\) | \(F_1(s)F_2(s)\) |
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