Basic Definitions and Results
Let f(t) be a function defined on
The domain of
- (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
whereis 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,
Solution: Since
we get
Hence,
In particular, we have
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