Laplace Transform

Definition

Assume that $f (t)$ is a functinon defined for $t \geq 0$ .
The Laplace Transform of $f$ is the function

F (s) = L {f} = \int_{0}^{\infty} e^{- s t} f (t) d t, s > a

Linearity of the Laplace Transform

Let $f (t)$ and $g (t)$ be functions that have a Laplace Transform that exists for $s > a$ , for some $a \in R$ .

Let $α, β \in R$ (or $C$ ) and denote $F (s) = L {f (t)}, G (s) = L {g (t)}$ .
Then

\begin{aligned} L {α f (t) + β g (t)} & = α L {f (t)} + β L {f (t)} & s > a \\ = α F (s) + β G (S) & s > a \end{aligned}

Assume $f (t)$ is piecewise continuous on $[0, \infty)$ , and of Exponential Order $α$ as $t \to \infty$ .
Then $L {f}$ exists for $x > α$
i.e.

F (x) = \int_{0}^{\infty} e^{- s t} f (t) d t = \underset{converges for x > α}{\underset{⏟}{lim_{N \to \infty} \int_{0}^{N} e^{- s t} f (t) d t}}

Given $F$ , assume $\exists$ another func $f$ such that $L {f} = F ⟹ f = L^{- 1} {F}$

$f (t) = L^{- 1} {f (s)}$