Hence,

(III.14)

Impulse function : See Fig. III.1(d) for the schematic of an unit impulse function. This is analogous to a pulse function whose duration is shrinked to zero without losing the strength. Hence the area under the impulse remains 1. The function can be expressed as the following:

(III.15)

As the duration of the impulse tends to zero, its maximum intensity ideally tends to . Mathematically it is termed as Dirac Delta function and is represented as . The following relation holds for unit impulse:

(III.16)

Thus the Laplace transform of the impulse function can be derived as the following:

(III.17)

L'Hospital's rule has been applied in the above derivation. Hence,

(III.18)

The following table presents the Laplace transforms of various functions.