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:
where
is an initial or boundary condition of
Proof:
We integrate by parts.
The Laplace transform transform second derivatives:
whereand
are initial or boundary conditions of
and
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 because
and 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:
If
then
Proof:
Changing the order of integration gives
Make the substitution
to give