Laplace Transform

Topics: Integral Transform

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(definition)

The Laplace transform is an integral transform defined as:

$$\mathcal{L}\{f(t)\}:={\int }_0^{\infty }{e}^{-st}f(t)dt=F(s)$$

We can easily obtain the Laplace transform various simple functions. We can prove several fundamental theorems for Laplace transforms.

We can easily obtain the Laplace transform of a piecewise function by first rewriting the piecewise function in terms of shifted unit step functions, then applying one of the previously mentioned fundamental theorems (see the corresponding note).

We can assure the existence of a Laplace transform through the consequences of various theorems.