Definition: $\displaystyle F(s) = \mathcal{L}{f(t)} = \int_0^\infty e^{-st}f(t)dt$
| f(t) | F(s) |
| $1$ | $\frac{1}{s}$ |
| $t^n$ | $\frac{n!}{s^{n + 1}}$ for $n = {n \in \mathbb{Z}: n > 0}$ |
| $e^{at}$ | $\frac{1}{s - a}$ |
| $sin(kt)$ | $\frac{k}{s^2 + k^2}$ |
| $cos(kt)$ | $\frac{s}{s^2 + k^2}$ |
| $\frac{d}{dt}$ | $sF(s) - f(0)$ |
| $\frac{d^2}{dt^2}$ | $s^2F(s) - sf(0) - f’(0)$ |
| $\frac{d^3}{dt^3}$ | $s^3F(s) - s^2f(0) - sf’(0) - f’‘(0)$ |
| $\int_0^t f(x)dx$ | $\frac{F(s)}{s}$ |
| f(t) | F(s) |
| $kf(t)$ | $kF(s)$ |
| $f_1(t) + f_2(t)$ | $F_1(s) + F_2(s)$ |
| $e^{at}f(t)$ | $F(s - a)$ |
| $f(t - a)U(t - a)$ | $e^{-as}F(s)$ for $a > 0$ |
| $t^nf(t)$ | $(-1)^n\frac{d^n}{ds^n}F(s)$ |
| $(f \ast g)(t)$ | $F(s)G(s)$ |