###
Linearity

The

Laplace transform is a

*linear operator*. To show this, let

denote a

linear combination of

signals and

,

where

and

are real or complex constants. Then we have

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

###
Differentiation

The

*differentiation theorem* for

Laplace transforms states that

where

, and

is any
differentiable function that approaches zero as

goes to infinity.
In operator notation,

*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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