Odel A Place Periodic
Odel A Place Periodic
Odel A Place Periodic
Recall that a function f (t) is said to be periodic of period T if f (t + T ) = f (t) for all t. The goal of this
handout is to prove the following (I even give two different proofs here).
Theorem 1. If f (t) is periodic with period T and piecewise continuous on the interval [0, T ], then the Laplace
transform F (s) of f (t) satisfies:
Z T
1
est f (t) dt
(1)
F (s) =
1 esT 0
Proof. Write the Laplace transform of f (t) as two integrals:
Z
(2)
f (t)e
F (s) =
st
f (t)e
ds =
st
f (t)est ds.
ds +
T
F (s) =
0
f (t)est ds
The claim is true but formally you have to justify it (here you have an infinite sum, so it does not formally
follow from the linearity of Laplace transform).
Further by the translation in t property
L{ukT (t)g(t kT )} = ekT s G(s),
where G(s) = L{g(t)}(s) =
(4)
RT
0
est f (t) dt
0
1
Now recall that 1 + x + x2 + x3 + ... = 1x
for |x| < 1 (the formula for the infinite sum of geometric
sT
2sT
3sT
progression). Therefore, (1 + e
+e
+e
+ . . .) = 1e1sT . Substituting this to (4) we get the required
formula (1).
1