Laplace transform
Properties and theorems

The Laplace transform's key property is that it converts differentiation and integration in the time domain into multiplication and division by s in the Laplace domain. Thus, the Laplace variable s is also known as operator variable in the Laplace domain: either the derivative operator or (for s) the integration operator.

Given the functions f (t) and g (t), and their respective Laplace transforms F (s) and G (s), f (t) = L − 1 { F } (s), g (t) = L − 1 { G } (s), {\displaystyle {\begin{aligned}f(t)&={\mathcal {L}}^{-1}\{F\}(s),\\g(t)&={\mathcal {L}}^{-1}\{G\}(s),\end{aligned}}}

the following table is a list of properties of unilateral Laplace transform:
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