AboutPosts
AboutResumeGallery
ToolsAdvanced SearchGPT-4o Prompts
ProjectsBB EnhancedDORM WIFIS-R-H-C

The Delta Function

SCU

Definition

∫pp+ϵδpδ(t)dt=1\int_p^{p+\epsilon}\delta_p^\delta(t)dt=1∫pp+ϵ​δpδ​(t)dt=1 δpϵ(t)={1ϵ, p≤t<p+ϵ0, elsewhere\delta_p^\epsilon(t)=\left\{\begin{aligned}\frac{1}{\epsilon},&\ p\le t\lt p+\epsilon\\0, &\ elsewhere\end{aligned}\right.δpϵ​(t)=⎩⎨⎧​ϵ1​,0,​ p≤t<p+ϵ elsewhere​ lim⁡ϵ→0δpϵ=δp\lim_{\epsilon\to 0}\delta_p^\epsilon=\delta_pϵ→0lim​δpϵ​=δp​

Combined with Laplace Transform

L{δp(t)}(s)=∫0+∞1ϵe−stdt=lim⁡ϵ→0∫pp+ϵ1ϵe−stdt=lim⁡ϵ→01ϵ∫pp+ϵe−stdt=lim⁡ϵ→01ϵ1(−s)e−st∣pp+ϵ=lim⁡ϵ→01−ϵs(e−s(p+ϵ)−e−sp)=lim⁡ϵ→01−ϵs(e−sϵ−1)e−sp=lim⁡ϵ→0e−sϵ−1−ϵse−sp=lim⁡t→0e−t−1−te−sp=lim⁡t→0(e−t−1)′−t′e−sp=lim⁡t→0−e−t−1e−sp=e−sp\begin{aligned}\mathcal{L}\{\delta_p(t)\}(s)&=\int_{0}^{+\infty}\frac{1}{\epsilon}e^{-st}dt\\&=\lim_{\epsilon\to 0}\int_{p}^{p+\epsilon}\frac{1}{\epsilon}e^{-st}dt\\&=\lim_{\epsilon\to 0}\frac1\epsilon\int_{p}^{p+\epsilon}e^{-st}dt\\&=\lim_{\epsilon\to 0}\frac1\epsilon \frac{1}{(-s)}e^{-st}|_{p}^{p+\epsilon}\\&=\lim_{\epsilon\to 0}\frac1{-\epsilon s}(e^{-s(p+\epsilon)}-e^{-sp})\\&=\lim_{\epsilon\to 0}\frac{1}{-\epsilon s}(e^{-s\epsilon}-1)e^{-sp}\\&=\lim_{\epsilon\to 0}\frac{e^{-s\epsilon}-1}{-\epsilon s}e^{-sp}\\&=\lim_{t\to0}\frac{e^{-t}-1}{-t}e^{-sp}\\&=\lim_{t\to 0}\frac{(e^{-t}-1)'}{-t'}e^{-sp}\\&=\lim_{t\to 0}\frac{-e^{-t}}{-1}e^{-sp}\\&=e^{-sp}\end{aligned}L{δp​(t)}(s)​=∫0+∞​ϵ1​e−stdt=ϵ→0lim​∫pp+ϵ​ϵ1​e−stdt=ϵ→0lim​ϵ1​∫pp+ϵ​e−stdt=ϵ→0lim​ϵ1​(−s)1​e−st∣pp+ϵ​=ϵ→0lim​−ϵs1​(e−s(p+ϵ)−e−sp)=ϵ→0lim​−ϵs1​(e−sϵ−1)e−sp=ϵ→0lim​−ϵse−sϵ−1​e−sp=t→0lim​−te−t−1​e−sp=t→0lim​−t′(e−t−1)′​e−sp=t→0lim​−1−e−t​e−sp=e−sp​