## Laplace transform

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
(ii) or
(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

and

From the last inequality we have