## Properties of the Laplace Transformation

The Laplace transform is linear:

This is actually a property of the integral, and is inherited by the Laplace transform.

The Laplace transform transforms first derivatives:

whereis an initial or boundary condition of

Proof:

We integrate by parts.

The Laplace transform transform second derivatives:

whereandare initial or boundary conditions ofand

Proof:

Again we integrate by parts (twice).

The Laplace transform transforms integrals:

Proof:

Again we integrate by parts.

At the upper limit the first term on the right vanishes becauseand at the lower limit the integral in the first term on the right is zero because the upper and lower limits are equal hence

The Laplace transform obeys the convolution principle:

Ifthen

Proof:

Changing the order of integration gives

Make the substitutionto give