常用函数的拉氏变换表

常用函数的拉氏变换表

拉氏变换
$$L[f(t)]=F(s)={\int }_0^{\infty }f(t)e^{-st}dt;(s+\sigma +j\omega 为复变量)$$

序号	原函数f(t)	象函数F(s)
1	$$\delta (t)$$	1
2	$$\epsilon (t)$$	$$\frac{1}{s}$$
3	t	$$\frac{1}{s^2}$$
4	$$\frac{t^{n-1}}{(n-1)!},n=1,2,...$$	$$\frac{1}{s^n}$$
5	$$e^{-at}$$	$$\frac{1}{s+a}$$
6	$$sin\omega t$$	$$\frac{\omega }{s^2+{\omega }^2}$$
7	$$cos\omega t$$	$$\frac{s}{s^2+{\omega }^2}$$
8	$$1-e^{-at}$$	$$\frac{a}{s(s+a)}$$
9	$$e^{-at}sin\omega t$$	$$\frac{\omega }{(s+a{)}^2+{\omega }^2}$$
10	$$e^{-at}cos\omega t$$	$$\frac{\omega +a}{(s+a{)}^2+{\omega }^2}$$
11	$$\begin{gathered} 1-\frac{1}{\sqrt{1-{\xi }^2}}{e}^{-\xi {\omega }_{n}t}sin(\sqrt{1-{\xi }^2}{\omega }_{n}t+\theta ) \\ \theta =arctan(\sqrt{1-{\xi }^2}/\xi ) \end{gathered}$$	$$\frac{{\omega }_n^2}{s(s^2+2\xi {\omega }_{n}s+{\omega }_n^2)}$$
12	$$\frac{{\omega }_n}{\sqrt{1-{\xi }^2}}{e}^{-\xi {\omega }_{n}t}sin(\sqrt{1-{\xi }^2}{\omega }_{n}t)$$	$$\frac{{\omega }_n^2}{s^2+2\xi {\omega }_{n}s+{\omega }_n^2}$$
13	$$\frac{1}{\beta -a}(e^{-at}-e^{-\beta t})$$	$$\frac{1}{(s+a)(s+\beta )}$$
14	$$\frac{1}{a^2}(e^{-at}+at-1)$$	$$\frac{1}{s^2(s+a)}$$
15	$$\frac{1}{(n-1)!}{t}^{n-1}{e}^{-at},n=1,2,...$$	$$\frac{1}{(s+a{)}^n}$$
16	$$\frac{1}{a^2}[1-(1+at)e^{-at}]$$	$$\frac{1}{s(s+a{)}^2}$$
17	$$\frac{1}{{\omega }^2}[1-cos(wt)]$$	$$\frac{1}{s(s^2+{\omega }^2)}$$
18	$$\begin{gathered} \frac{a_0}{({\omega }_0}-\frac{a_0^2+{\omega }^2{)}^{1/2}}{{\omega }^2}cos(\omega t+\psi ) \\ \psi =arctan(\omega /a_0) \end{gathered}$$	$$\frac{s+a_0}{s(s^2+{\omega }^2)}$$
19	$$\frac{1}{ab}+\frac{1}{ab(a-b)}(b{e}^{-at}-a{e}^{-bt})$$	$$\frac{1}{s(s+a)(s+b)}$$
20	$$\frac{a_0}{ab}+\frac{a_0-a}{a(a-b)}{e}^{-at}+\frac{a_0-b}{b(b-a)}{e}^{-bt})$$	$$\frac{s+a_0}{s(s+a)(s+b)}$$
21	$$\frac{a_0}{ab}+\frac{a^2-a_1-a_0}{a(a-b)}{e}^{-at}+\frac{b^2-a_{1}b+a_0}{b(a-b)}{e}^{-bt})$$	$$\frac{s+a_{1}s+a_0}{s(s+a)(s+b)}$$