ONE-SIDED LAPLACE TRANSFORM!

The Laplace transform is a major conceptual and analytical tool of system theory, and hence we explore its properties in more detail below. Table 2.3 lists the Laplace transforms of a number of commonly occurring functions. It should be noted that (i) the value of the function for t < 0 is not relevant to x(s) and (ii) that the integral (2.3,7) may diverge for some x(t) in combi­nation with some values of s, in which case x(s) does not exist. This re­striction is weak, and excludes few cases of interest to engineers, (iii) When the function is zero for t < 0, the Fourier transform is obtained from the Laplace transform by replacing s by ia>.

TRANSFORMS OF DERIVATIVES

Given the function x(t), the transforms of its derivatives can be found from (2.3,7).

When xe~st —> 0 as t —>• oo (only this case is considered), then

= -*(0) + Ще) (2.3,14)

where x(0) is the value of x(t) when f = 0.J The process may be repeated to find the higher derivatives by replacing x(t) in (2.3,14) by x(t), and so on. The result is

f In the two-sided Laplace transform, the lower limit of the integral is — со instead of zero.

J To avoid ambiguity when dealing with step functions, t = 0 should always be interpreted as t = 0+.