User:Vanished user fijw983kjaslkekfhj45/Laplace Transform Tables

Definition

 * $$F(s) = \mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt. $$

The parameter s is a complex number:


 * $$s = \sigma + i \omega, \, $$ with real numbers σ and ω.

Inverse Laplace transform

 * $$f(t) = \mathcal{L}^{-1} \{F(s)\} = \frac{1}{2 \pi i} \lim_{T\to\infty}\int_{ \gamma - i T}^{ \gamma + i T} e^{st} F(s)\,ds,$$

where $$\gamma$$ is a real number so that the contour path of integration is in the region of convergence of F(s).

Properties and theorems

 * Initial value theorem:
 * $$f(0^+)=\lim_{s\to \infty}{sF(s)}.$$


 * Final value theorem:
 * $$f(\infty)=\lim_{s\to 0}{sF(s)}$$, if all poles of $$ sF(s) $$ are in the left half-plane.