## Laplace Transform Theorems

### Linearity

The Laplace transform is a *linear operator*. To show this, let
denote a linear combination of signals and ,

Thus, linearity of the Laplace transform follows immediately from the linearity of integration.

### Differentiation

The *differentiation theorem* for Laplace transforms states that

*Proof: *
This follows immediately from integration by parts:

since by assumption.

**Corollary: ***Integration Theorem*

Thus, successive time derivatives correspond to successively higher powers of , and successive integrals with respect to time correspond to successively higher powers of .

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