Definition:Laplace Transform/Discontinuity at Zero

Definition
Let $f: \R_{> 0} \to \mathbb F$ be a function of a real variable $t$, where $\mathbb F \in \set {\R, \C}$.

Let $f$ be discontinuous or not defined at $t = 0$.

Then the Laplace transform of $f$ is defined as:


 * $\ds \laptrans {\map f t} = \map F s = \int_{0^+}^{\to +\infty} e^{-s t} \map f t \rd t = \lim_{\epsilon \mathop \to 0^+} \int_\epsilon^{\to +\infty} e^{-s t} \map f t \rd t$

whenever this improper integral converges.

If this improper integral does not converge, then $\laptrans {\map f t}$ does not exist.

Here the integral is improper not only because of its upper limit but also because of its lower limit.