DEFINITION of Laplace Transfrom:

\(\mathscr{L}\{x(t)\} = \int_{0}^{\infty}{x(t)e^{-st}dt} = \lim\limits_{M \to \infty} \int_{0}^{M}{x(t)e^{-st}dt}\\\) An integral is transformable with a Laplace transform only if the integral is Convergent

DEFINITION of a function with Exponential growth:

\(x:\mathbb{R} \mapsto \mathbb{R}\\ t \mapsto x(t)\\ \text{If } \exists M > 0\\ \exists\gamma \in \mathbb{R} \text{ such that} \begin{vmatrix} x(t) \end{vmatrix}\le Me^{\gamma t}, \forall t \ge 0\\\) and we define the Order of Growth

DEFINITION piecewise continuous functions:

\(x: \mathbb{R} \mapsto \mathbb{R}\) is piecewise Continuous
If in every closed and bounded interval it has at most a finite number of singularities of the following kinds:

• $1^{\text{st}}$ Kind (Jump)
• $3^{\text{rd}}$ Kind (Removable)

Theorem:

Initial Value theorem \(x \in \mathscr{L}(0, +\infty) \implies \lim\limits_{t \to 0^{+}}X(t) = \lim\limits_{s \to +\infty}s X(s)\)

Final Value Theorem \(x \in \mathscr{L}(0, +\infty) \implies \lim\limits_{t \to 0^{+}}X(t) = \lim\limits_{s \to +\infty}s X(s)\)