Begin2.DVI

Figure 12-9. Inverting the Laplace transform.

Properties of the Laplace transform

Using the definition of the Laplace transform and applying various integration

techniques one can develop a table of Laplace transform properties.

Example 12-13. (Laplace transform of derivative)

Show that if L{f(t)}=F(s)then

L{ f′(t)}=sF (s)−f(0 +)or f′(t) = L−^1

{

sF (s)−f(0 +)

}

Solution

By definition

L{ f′(t)}=

∫∞

0

f′(t)e−st dt

Integrate by parts with u=e−st and dv =f′(t)dt to obtain

L{ f′(t)}=e−stf(t)

∞

0

+s

∫∞

0

f(t)e−st dt =sF (s)−f(0 +)

Note that f(t)need only be defined for t≥ 0 so that the 0 +is to remind you that the

function f(t)evaluated at zero is just the right-hand limit as t→ 0.