Laplace transform

Table of Laplace transforms

Specific functions

Laplace transform	Function
$1 (𝑠 - 𝑎) 𝑛$	$𝑒 𝑎 𝑡 𝑡 𝑛 - 1 (𝑛 - 1)!$
$1 𝑠 2 + 𝜔 2$	$s i n (𝜔 𝑡) 𝜔$
$𝑠 𝑠 2 + 𝜔 2$	$c o s (𝜔 𝑡)$
$1 (𝑠 - 𝑎) 2 + 𝜔 2$	$𝑒 𝑎 𝑡 s i n (𝜔 𝑡) 𝜔$
$𝑠 - 𝑎 (𝑠 - 𝑎) 2 + 𝜔$	$𝑒 𝑎 𝑡 c o s (𝜔 𝑡)$
$1 (𝑠 2 + 𝜔 2) 2$	$s i n (𝜔 𝑡) - 𝜔 𝑡 c o s (𝜔 𝑡) 2 𝜔 3$
$𝑠 (𝑠 2 + 𝜔 2) 2$	$𝑡 s i n (𝜔 𝑡) 2 𝜔$

Note $𝑛 \in ℕ \leq 0$

General rules

Laplace transform	Function
$𝑒 - 𝑎 𝑠 𝑠$	$𝐻 (𝑡 - 𝑎)$
$𝑒 - 𝑎 𝑠 \cdot L {𝑓} (𝑠)$	$𝑓 (𝑡 - 𝑎) 𝐻 (𝑡 - 𝑎)$
$L {𝑓} (𝑠 - 𝑎)$	$𝑒 𝑎 𝑡 𝑓 (𝑡)$
$𝑠 L {𝑓} (𝑠) - 𝑓 (0)$	$𝑓' (𝑡))$
$𝑠 2 L {𝑓} (𝑠) - 𝑠 𝑓 (0) - 𝑓' (0)$	$𝑓 ″ (𝑡)$
$(𝐷 L) {𝑓} (𝑠)$	$- 𝑡 𝑓 (𝑡)$
$(𝐷 𝑛 L) {𝑓} (𝑠)$	$(- 𝑡) 𝑛 𝑓 (𝑡)$
$L {𝑓} (𝑠) 𝑠$	$\int 𝑡 0 𝑓 (𝑢) 𝑑 𝑢$
$L {𝑓} (𝑠) L {𝑔} (𝑠)$	$(𝑓 * 𝑔) (𝑡)$

Note that here $𝐻 (𝑡)$ represents the Heaviside function and $𝑓 * 𝑔$ represents Convolution. $𝐷$ is the differential operator.

#state/tidy | #SemBr