In this topic, you study the Table of Laplace Transforms.
Definition: Laplace transform of $f(t)$ is
\[L[f(t)] = F(s) = \int\limits_0^\infty {f(t){e^{ – st}}dt}\]
Using above property, the Laplace transform of Basic Functions are
| \[L[f(t)] \] | \[F(s)\] |
| \[L[u(t)] \] | \[ \frac{1}{s} \] |
| \[L[t] \] | \[ \frac{1}{{{s^2}}} \] |
| \[L[{t^n}] \] | \[ \frac{{n!}}{{{s^{n + 1}}}} \] |
| \[L[{e^{ – at}}] \] | \[\frac{1}{{s+a}} \] |
| \[L[e^{ at}] \] | \[\frac{1}{{s-a}} \] |
| \[L[\sin at] \] | \[ \frac{{a}}{{{s^2} + a^2}} \] |
| \[L[\cos at] \] | \[ \frac{{s}}{{{s^2} + a^2}} \] |
| \[L[\sinh at] \] | \[ \frac{{a}}{{{s^2} – a^2}} \] |
| \[L[\cosh at] \] | \[ \frac{{s}}{{{s^2} – a^2}} \] |
Note:- Defined for $t$ ≥ 0, $f(t)$ = 0, for $t$ < 0.