Laplace Final Value Theorem Proof
$\Begingroup$ I Was Studing Fvt and the Proof I Used Was This: $$\Int_0^\Infty \Hat F(T)E^{-St}\, Dt=Sf(S)-F(0)$$ $$\Lim_{S\To0}\Int_0^\Infty \Hat...
I was studing FVT and the proof i used was this : $$\int_0^\infty \hat f(t)e^{-st}\,dt=sF(s)-f(0)$$ $$\lim_{s\to0}\int_0^\infty \hat f(t)e^{-st}\,dt=\lim_{s\to0}[sF(s)]-f(0)$$ $$\int_0^\infty \hat f(t)\,dt=\lim_{s\to0}[sF(s)]-f(0)$$ $$f(\infty)-f(0)=\lim_{s\to0}[sF(s)]-f(0)$$ $$f(\infty)=\lim_{s\to0}[sF(s)]$$ it looks straightforward, but i've heard about a condition : " The standard assumptions for the final value theorem require that the Laplace transform have all of its poles either in the open-left-half plane (OLHP) or at the origin, with at most a single pole at the origin. In this case, the time function has a finite limit." now i'm just wondering regarding the proof where does this come from ? why is this condition needed? What is the reason behind it ?
Maya is a hardware enthusiast who tests and reviews smart home devices, smartphones, wearables, and audio gear. She focuses on practical consumer value and build quality.
