# Transform of the Unit-Impulse Function

Consider the following example.

EXAMPLE 10.6.1   Find the Laplace transform, , of

Solution: Note that By linearity of the Laplace transform, we get

Remark 10.6.2
1. Observe that in Example 10.6.1, if we allow to approach 0 , we obtain a new function, say That is, let

This new function is zero everywhere except at the origin. At origin, this function tends to infinity. In other words, the graph of the function appears as a line of infinite height at the origin. This new function, , is called the UNIT-IMPULSE FUNCTION (or Dirac's delta function).

2. We can also write

3. In the strict mathematical sense does not exist. Hence, mathematically speaking, does not represent a function.

4. However, note that

5. Also, observe that Now, if we take the limit of both sides, as approaches zero (apply L'Hospital's rule), we get

A K Lal 2007-09-12