## Laplace Transform Theorems

### 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 .
Next Section:
Laplace Analysis of Linear Systems
Previous Section:
Relation to the z Transform