Math 555: Differential Equations
Table of Laplace Transforms
| $\Large f(t) = \mathcal{L}^{-1}\{F(s)\}$ | $\Large F(s) = \mathcal{L}\{f(t)\}$ | |
|---|---|---|
| 1. | $\large 1$ | $\large \dfrac{1}{s},\quad s > 0$ |
| 2. | $\large e^{at}$ | $\large \dfrac{1}{s-a},\quad s > a$ |
| 3. | $\large t^n,\quad n\in\mathbb{N}$ | $\large \dfrac{n!}{s^{n+1}},\quad s > 0$ |
| 4. | $\large t^p,\quad p\gt -1$ | $\large \dfrac{\Gamma(p+1)}{s^{p+1}},\quad s > 0$ |
| 5. | $\large \sin(at)$ | $\large \dfrac{a}{s^2+a^2},\quad s > 0$ |
| 6. | $\large \cos(at)$ | $\large \dfrac{s}{s^2+a^2},\quad s > 0$ |
| 7. | $\large \sinh(at)$ | $\large \dfrac{a}{s^2-a^2},\quad s > \vert a\vert$ |
| 8. | $\large \cosh(at)$ | $\large \dfrac{s}{s^2-a^2},\quad s > \vert a\vert$ |
| 9. | $\large e^{at}\sin(bt)$ | $\large \dfrac{b}{(s-a)^2 + b^2},\quad s > a$ |
| 10. | $\large e^{at}\cos(bt)$ | $\large \dfrac{s-a}{(s-a)^2 + b^2},\quad s > a$ |
| 11. | $\large t^n e^{at},\quad n\in\mathbb{N}$ | $\large \dfrac{n!}{(s-a)^{n+1}},\quad s > a$ |
| 12. | $\large u_c(t) = \left\{\begin{matrix} 0 & t\lt c \\ 1 & t\ge c \end{matrix}\right.$ | $\large \dfrac{e^{-cs}}{s},\quad s > 0$ |
| 13. | $\large u_c(t)f(t-c)$ | $\large e^{-cs}F(s)$ |
| 14. | $\large e^{ct}f(t)$ | $\large F(s-c)$ |
| 15. | $\large f(ct)$ | $\large \dfrac{1}{c}F\left(\dfrac{s}{c}\right),\quad c > 0$ |
| 16. | $\large (f*g)(t) = \int_0^t f(t-\tau)g(\tau)\,d\tau$ | $\large F(s)G(s)$ |
| 17. | $\large \delta(t-c)$ | $\large e^{-cs}$ |
| 18. | $\large f^{(n)}(t)$ | $\large s^n F(s) - s^{n-1}f(0) -\ldots - f^{(n-1)}(0)$ |
| 19. | $\large (-t)^n f(t)$ | $\large F^{(n)}(s)$ |