表 1-1 常用函数拉普拉斯变换对照表
| 序号 | 象函数 \(F(s)\) | 原函数 \(f(t)\) |
|---|---|---|
| 1 | \(1\) | \(\delta(t)\) |
| 2 | \(\dfrac{1}{s}\) | \(1(t)\) |
| 3 | \(\dfrac{1}{s^2}\) | \(t\) |
| 4 | \(\dfrac{1}{s^n}\) | \(\dfrac{t^{n-1}}{(n-1)!}\) |
| 5 | \(\dfrac{1}{s+a}\) | \(e^{-at}\) |
| 6 | \(\dfrac{1}{(s+a)(s+b)}\) | \(\dfrac{1}{(b-a)}(e^{-at}-e^{-bt})\) |
| 7 | \(\dfrac{s+a_0}{(s+a)(s+b)}\) | \(\dfrac{1}{(b-a)}[(a_0-a)e^{-at}-(a_0-b)e^{-bt}]\) |
| 8 | \(\dfrac{1}{s(s+a)(s+b)}\) | \(\dfrac{1}{ab}+\dfrac{1}{ab(a-b)}[be^{-at}-ae^{-bt}]\) |
| 9 | \(\dfrac{s+a_0}{s(s+a)(s+b)}\) | \(\dfrac{a_0}{ab}+\dfrac{a_0-a}{a(a-b)}e^{-at}+\dfrac{a_0-b}{b(b-a)}e^{-bt}\) |
| 10 | \(\dfrac{s^2+a_1s+a_0}{s(s+a)(s+b)}\) | \(\dfrac{a_0}{ab}+\dfrac{a^2-a_1a+a_0}{a(a-b)}e^{-at}-\dfrac{b^2-a_1b+a_0}{b(a-b)}e^{-bt}\) |
| 11 | \(\dfrac{1}{(s+a)(s+b)(s+c)}\) | \(\dfrac{e^{-at}}{(b-a)(c-a)}+\dfrac{e^{-bt}}{(a-b)(c-b)}+\dfrac{e^{-ct}}{(a-c)(b-c)}\) |
| 12 | \(\dfrac{s+a_0}{(s+a)(s+b)(s+c)}\) | \(\dfrac{a_0-a}{(b-a)(c-a)}e^{-at}+\dfrac{a_0-b}{(a-b)(c-b)}e^{-bt}+\dfrac{a_0-c}{(a-c)(b-c)}e^{-ct}\) |
| 13 | \(\dfrac{s^2+a_1s+a_0}{(s+a)(s+b)(s+c)}\) | \(\dfrac{a^2-a_1a+a_0}{(b-a)(c-a)}e^{-at}+\dfrac{b^2-a_1b+a_0}{(a-b)(c-b)}e^{-bt}+\dfrac{c^2-a_1c+a_0}{(a-c)(b-c)}e^{-ct}\) |
| 14 | \(\dfrac{1}{s^2+\omega^2}\) | \(\dfrac{1}{\omega}\sin\omega t\) |
| 15 | \(\dfrac{s}{s^2+\omega^2}\) | \(\cos\omega t\) |
| 16 | \(\dfrac{s+a_0}{s^2+\omega^2}\) | \(\dfrac{1}{\omega}(a_0^2+\omega^2)^{1/2}\sin(\omega t+\varphi)\), \(\varphi=\arctan\dfrac{\omega}{a_0}\) |
| 17 | \(\dfrac{1}{s(s^2+\omega^2)}\) | \(\dfrac{1}{\omega^2}(1-\cos\omega t)\) |
| 18 | \(\dfrac{s+a_0}{s(s^2+\omega^2)}\) | \(\dfrac{a_0}{\omega^2}-\dfrac{(a_0^2+\omega^2)^{1/2}}{\omega^2}\cos(\omega t+\varphi)\), \(\varphi=\arctan\dfrac{\omega}{a_0}\) |
| 19 | \(\dfrac{s+a_0}{(s+a)(s^2+\omega^2)}\) | \(\dfrac{a_0-a}{a^2+\omega^2}e^{-at}+\dfrac{1}{\omega}\left(\dfrac{a_0^2+\omega^2}{a^2+\omega^2}\right)^{1/2}\sin(\omega t+\varphi)\) \(\varphi=\arctan\dfrac{\omega}{a_0}-\arctan\dfrac{\omega}{a}\) |
| 20 | \(\dfrac{1}{(s+a)^2+\omega^2}\) | \(\dfrac{1}{\omega}e^{-at}\sin\omega t\) |
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