The Delta Function

Definition

$${\int }_p^{p+ϵ}{\delta }_p^{\delta }(t)dt=1$$$${\delta }_p^{ϵ}(t)=\{\begin{array}{rl}\frac{1}{ϵ},& p\le t<p+ϵ\\ 0,& elsewhere\end{array}$$$$\underset{ϵ\to 0}{lim}{\delta }_p^{ϵ}={\delta }_p$$

Combined with Laplace Transform

$$\begin{array}{rl}\mathcal{L}\{{\delta }_p(t)\}(s)& ={\int }_0^{+\mathrm{\infty }}\frac{1}{ϵ}{e}^{-st}dt\\ & =\underset{ϵ\to 0}{lim}{\int }_p^{p+ϵ}\frac{1}{ϵ}{e}^{-st}dt\\ & =\underset{ϵ\to 0}{lim}\frac{1}{ϵ}{\int }_p^{p+ϵ}{e}^{-st}dt\\ & =\underset{ϵ\to 0}{lim}\frac{1}{ϵ}\frac{1}{(-s)}{e}^{-st}{|}_p^{p+ϵ}\\ & =\underset{ϵ\to 0}{lim}\frac{1}{-ϵs}(e^{-s(p+ϵ)}-e^{-sp})\\ & =\underset{ϵ\to 0}{lim}\frac{1}{-ϵs}(e^{-sϵ}-1)e^{-sp}\\ & =\underset{ϵ\to 0}{lim}\frac{e^{-sϵ}-1}{-ϵs}{e}^{-sp}\\ & =\underset{t\to 0}{lim}\frac{e^{-t}-1}{-t}{e}^{-sp}\\ & =\underset{t\to 0}{lim}\frac{(e^{-t}-1{)}^{\prime }}{-t^{\prime }}{e}^{-sp}\\ & =\underset{t\to 0}{lim}\frac{-e^{-t}}{-1}{e}^{-sp}\\ & =e^{-sp}\end{array}$$