Let be a function on . The Laplace transform of is defined by the integral
The domain of is all values of such that the integral exists.
Let and be functions whose Laplace transforms exist for and let and be constants. Then, for ,
which can be proved using the properties of improper integrals.
If the Laplace transform exists for , then
for .
Proof.
Laplace Transform of Higher-Order Derivatives[edit | edit source]
If , then
- Proof:
- (integrating by parts)
Using the above and the linearity of Laplace Transforms, it is easy to prove that
Derivatives of the Laplace Transform[edit | edit source]
If , then
Laplace Transform of Few Simple Functions[edit | edit source]