Lecture Notes in Differential Equations - Bruce E. Shapiro

355
Similarly, we can express a unit impulse centered at t 0 by a right shift, as
δ a (t − t 0 ), and an infinite unite spike as δ(t − t 0 ).
Figure 32.6: Unit impulse of width a at the origin (left). Sequence of
sharper and sharper impulses on the right, with smaller and smaller a
values.
1 2a
5
a
a
4
3
2
1
0
2 1 0 1 2
The Laplace transform of the unit impulse is
[ ]
1
L[δ a (t − t 0 )] = L
2a (U(t + a − t 0) − U(t − a − t 0 ))
(32.222)
= 1
2a [L[U(t + a − t 0)] − L[U(t − a − t 0 )]] (32.223)
= e(a−t0)s
2as
− e−(a+t0)s
2as
= e −t0s eas − e −as
2as
−t0s sinh as
= e
as
(32.224)
(32.225)
(32.226)
To get the Laplace transform of the delta function we take the limit as
a → 0,
−t0s sinh as
L[δ(t − t 0 )] = lim e
a→0 as
(32.227)
Since the right hand side → 0/0 we can use L‘Hopital’s rule from calculus,
L[δ(t − t 0 )] = e −t0s a cosh as
lim
a→0 a
(32.228)
= e −t0s lim cosh as
a→0
(32.229)
= e −t0s (32.230)