Convolution Product

One of the most common mistake students will commit is

Although it is tempting to assume that this is true, one may easily check that it is wrong by taking f(t) = t, and g(t) = t. Therefore, a natural question is: what would  be equal to ?
The answer to this question is given by the formula

where
,
is called the convolution product of f(t), and g(t). The following are some of the basic properties of the convolution product:

(1)

 ;

(2)

 ;

(3)

;

(4)

 ;

(5)

.

Example: Find the solution to

Solution: Apply the Laplace transform to get

where  . Hence,

We then rewrite Y(s) to get

Therefore, we have