Laplace transform is a powerful tool which allows reducing relations between functions and its derivatives to some algebraic relations. Of course , former statement gives just a little idea about the power of Laplace transform.
Laplace transform is a correspondence which associate to a measurable function with a certain growth at infinite a complex olomorph function in a semi space in the complex plan.
Let a function with properties:
(i)f is a measurable function
(iii)there is a couple or real constants such as
For such a function we take a complex with and define
The function is in the Lebesgue space and using theorem of differentiability of integral with parameter we have, as
This is valid for as for such a s we have
From the last inequality we have