Lecture Notes in Differential Equations - Bruce E. Shapiro

352 LESSON 32. THE LAPLACE TRANSFORM
Substituting back into (32.186),
f(t) =
(− 1 6 e−5t + 1 ) ( 5
6 et 6 e5t + 1 )
6 et
(32.193)
Periodic Functions
Definition 32.14. A function f(t) is said to be periodic with period T
if there exists some positive number such that
for all values of t.
f(t + T ) = f(t) (32.194)
To find the transform of a periodic function we only have to integrate over
a single interval, as illustrated below. Let f(t) be a periodic function with
period T . Then its transform is
F (S) =
=
∫ ∞
0
∫ t
0
f(t)e −ts dt (32.195)
f(t)e −st dt +
∫ ∞
T
f(t)e −ts dt (32.196)
In the second integral make the substitution u = t − T , so that
F (S) =
=
=
∫ t
0
∫ t
0
∫ t
0
f(t)e −st dt +
∫ ∞
0
∫ ∞
f(t)e −st dt + e −sT f(u)e −us du
f(u + T )e −(u+T )s du (32.197)
0
since f(u + T ) = f(u)
(32.198)
f(t)e −st dt + e −sT F (s) (32.199)
Rearranging and solving for F (s),
F (s)(1 − e −sT ) =
∫ t
which gives us the following result for periodic functions:
0
f(t)e −st dt (32.200)
∫
1 t
F (s) = L[f(t)] =
1 − e −sT f(t)e −st dt (32.201)
0