Use Definition 711 DEFINITION 711 Laplace Transform Let f be

Use Definition 7.1.1.

DEFINITION 7.1.1    Laplace Transform
Let f be a function defined for

t 0.

is said to be the Laplace transform of f, provided that the integral converges.

Find

f(t) =

	estf(t) dt
0

Solution

F(s) = (from 0 to ) f(t) e^(-st) dt =
= (from 0 to 1) -e^(-st) dt + (from 1 to ) e^(-st) dt =
= (1/s) (e^(-s) - 1) + (1/s) e^(-s), if real(s) > 0, otherwise the second integral diverges.
=>
F(s) = (2 e^(-s) -1) / s

Other method:

h(t) = { 1 if 0 t;
{ 0 if t < 0.

f(t) = -h(t) + 2 h(t-1)

H(s) = 1/s (if real(s)>0), and L{f(t-)} = e^(-s) L{f(t)} if f(t) = 0 for t<0.
=>
F(s) = -1/s + 2 e(-s)/s