###
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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