Laplace transform

Table of Laplace transforms

Specific functions

Laplace transform	Function
1(𝑠−𝑎)𝑛
𝑒𝑎𝑡𝑡𝑛−1(𝑛−1)!
1𝑠2+𝜔2
sin⁡(𝜔𝑡)𝜔
𝑠𝑠2+𝜔2
cos⁡(𝜔𝑡)
1(𝑠−𝑎)2+𝜔2
𝑒𝑎𝑡sin⁡(𝜔𝑡)𝜔
𝑠−𝑎(𝑠−𝑎)2+𝜔
𝑒𝑎𝑡cos⁡(𝜔𝑡)
1(𝑠2+𝜔2)2
sin⁡(𝜔𝑡)−𝜔𝑡cos⁡(𝜔𝑡)2𝜔3
𝑠(𝑠2+𝜔2)2
𝑡sin⁡(𝜔𝑡)2𝜔

Note 𝑛 ∈ℕ≤0

General rules

Laplace transform	Function
𝑒−𝑎𝑠𝑠
𝐻(𝑡 −𝑎)
𝑒−𝑎𝑠 ⋅L{𝑓}(𝑠)
𝑓(𝑡 −𝑎) 𝐻(𝑡 −𝑎)
L{𝑓}(𝑠 −𝑎)
𝑒𝑎𝑡𝑓(𝑡)
𝑠L{𝑓}(𝑠) −𝑓(0)
𝑓′(𝑡))
𝑠2L{𝑓}(𝑠) −𝑠𝑓(0) −𝑓′(0)
𝑓″(𝑡)
(𝐷L){𝑓}(𝑠)
−𝑡𝑓(𝑡)
(𝐷𝑛L){𝑓}(𝑠)
( −𝑡)𝑛𝑓(𝑡)
L{𝑓}(𝑠)𝑠

$$ | | $$\mathcal{L}\{ f \}(s) \, \mathcal{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