# Common Laplace Transforms

Definition: $\displaystyle F(s) = \mathcal{L}{f(t)} = \int_0^\infty e^{-st}f(t)dt$

## Common Transforms

f(t) F(s)
$1$ $\frac{1}{s}$
$t^n$ $\frac{n!}{s^{n + 1}}$ for $n = {n \in \mathbb{Z}: n > 0}$
$e^{at}$ $\frac{1}{s - a}$
$sin(kt)$ $\frac{k}{s^2 + k^2}$
$cos(kt)$ $\frac{s}{s^2 + k^2}$
$\frac{d}{dt}$ $sF(s) - f(0)$
$\frac{d^2}{dt^2}$ $s^2F(s) - sf(0) - f’(0)$
$\frac{d^3}{dt^3}$ $s^3F(s) - s^2f(0) - sf’(0) - f’‘(0)$
$\int_0^t f(x)dx$ $\frac{F(s)}{s}$

## Common Operational Transforms

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 $a > 0$
$t^nf(t)$ $(-1)^n\frac{d^n}{ds^n}F(s)$
$(f \ast g)(t)$ $F(s)G(s)$