Definition and Existence Proof of the Laplace Transform
Definition[^1]
The Laplace transform of a function $f$ is defined as follows.
$$ \mathcal{L} \left\{ f(t) \right\} := \int _{0}^\infty e^{-st}f(t) dt =F(s) $$
The Laplace transform of a function $f$ is defined as follows.
$$ \mathcal{L} \left\{ f(t) \right\} := \int _{0}^\infty e^{-st}f(t) dt =F(s) $$