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78 CHAPTER 3 The Laplace Transform

and so on. If we include the variable, these would be written
L[ f ](s) = F(s), L[g](s) = G(s), and L[h](s) = H(s)
It is also customary to use t (for time) as the variable of the input function and s for the variable
of the transformed function.

EXAMPLE 3.1
at
Let a be any real number, and f (t) = e . The Laplace transform of f is the function defined by
∞

e dt
L[ f ](s) = e −st at
0
∞
k
= e (a−s)t dt = lim e (a−s)t dt
k→∞
0 0
k
1

= lim e (a−s)t
k→∞ a − s
0
1 1
=− =
a − s s − a
at
provided that s > a. The Laplace transform of f (t) = e can be denoted F(s) = 1/(s − a) for
s > a.
We rarely determine a Laplace transform by integration. Table 3.1 is a short table of Laplace
transforms of familiar functions, and much longer tables are available. In this table, n denotes a
nonnegative integer, and a and b are constants. Reading from the table (left to right), if f (t) =
sin(3t) then by entry (6), we have
3
F(s) = ,
2
s + 9
2t
and if k(t) = e cos(5t) then by entry (11), we have
s − 2
K(s) = .
(s − 2) + 25
2
There are also software routines for transforming functions. In MAPLE, first enter
with(inttrans);

TABLE 3.1 Laplace Transforms of Selected Functions

f (t) F(s) f (t) F(s)
1 2as
(1) 1 (8) t sin(at)
2
s (s + a )
2 2
2
n! s − a 2
(2) t n (9) t cos(at)
2 2
2
s n+1 (s + a )
1 b
at
(3) e at (10) e sin(bt)
s − a (s − a) + b 2
2
n! s − a
n at
(4) t e (11) e cos(bt)
at
(s − a) n+1 (s − a) + b 2
2
a − b a
at
(5) e − e bt (12) sinh(at)
(s − a)(s − b) s − a 2
2
a s
(6) sin(at) (13) cosh(at)
s + a 2 s − a 2
2
2
s
(7) cos(at) (14) δ(t − a) e −as
s + a 2
2
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October 14, 2010 14:14 THM/NEIL Page-78 27410_03_ch03_p77-120