## Definition of Laplace Transform

Let f(t) be a given function which is defined for t ≥ 0. If there exists a function F(s) so that

$\displaystyle F(s) = \int_0^\infty e^{-st} \, f(t) \, dt$,

then F(s) is called the Laplace Transform of f(t), and will be denoted by $\mathcal{L} \left\{f(t)\right\}$. Notice the integrator   e-st dt where s is a parameter which may be real or complex.

Thus,

$\mathcal{L} \left\{f(t)\right\} = F(s)$

The symbol $\mathcal{L}$ which transform f(t) into F(s) is called the Laplace transform operator.