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# Basic Definitions and Results

Let f(t) be a function defined on  . The Laplace transform of f(t) is a new function defined as

The domain of  is the set of , such that the improper integral converges.
(1)
We will say that the function f(t) has an exponential order at infinity if, and only if, there exist  and M such that
(2)
Existence of Laplace transform
Let f(t) be a function piecewise continuous on [0,A] (for every A>0) and have an exponential order at infinity with  . Then, the Laplace transform  is defined for  , that is .
(3)
Uniqueness of Laplace transform
Let f(t), and g(t), be two functions piecewise continuous with an exponential order at infinity. Assume that
then f(t)=g(t) for  , for every B > 0, except maybe for a finite set of points.
(4)
If  , then
(5)
Suppose that f(t), and its derivatives  , for  , are piecewise continuous and have an exponential order at infinity. Then we have
This is a very important formula because of its use in differential equations.
(6)
Let f(t) be a function piecewise continuous on [0,A] (for every A>0) and have an exponential order at infinity. Then we have
where  is the derivative of order n of the function F.
(7)
Let f(t) be a function piecewise continuous on [0,A] (for every A>0) and have an exponential order at infinity. Suppose that the limit , is finite. Then we have
(8)
Heaviside function
The function
is called the Heaviside function at c. It plays a major role when discontinuous functions are involved. We have

When c=0, we write  . The notation , is also used to denote the Heaviside function.
(9)
Let f(t) be a function which has a Laplace transform. Then,
and

Hence,

Example: Find
.
Solution: Since
,
we get

Hence,

In particular, we have