LAPLACE TRANSFORM

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

The next example deals with the Gamma Function.

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