definition
The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), defined by:
- $ F(s) = \mathcal{L} \left\{f(t)\right\}=\int_{0^-}^{\infty} e^{-st} f(t) \,dt. $
The lower limit of 0− is short notation to mean
- $ \lim_{\varepsilon\to 0+}\int_{-\varepsilon}^\infty $
and assures the inclusion of the entire Dirac delta function δ(t) at 0 if there is such an impulse in f(t) at 0.
The parameter s is in general complex number:
- $ s = \sigma + i \omega \, $
inverse laplace transform
The inverse Laplace transform is given by the following complex integral
- $ f(t) = \mathcal{L}^{-1} \{F(s)\} = \frac{1}{2 \pi i} \int_{ \gamma - i \cdot \infty}^{ \gamma + i \cdot \infty} e^{st} F(s)\,ds, $
