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Laplace transform of cosine and sine∗
pahio†
2013-03-22 1:32:01
We start from the easily derivable formula
eαt x
1
s−α
(s > α),
(1)
where the curved arrow points from the Laplace-transformed function to the
original function. Replacing α by −α we can write the second formula
e−αt x
1
s+α
(s > −α).
(2)
Adding (1) and (2) and dividing by 2 we obtain (remembering the linearity of
the Laplace transform)
1
1
eαt +e−αt
1
x
+
,
2
2 s−α s+α
i.e.
L{cosh αt} =
s
.
s2 −α2
(3)
Similarly, subtracting (1) and (2) and dividing by 2 give
L{sinh αt} =
a
.
s2 −α2
(4)
The formulae (3) and (4) are valid for s > |α|.
There are the hyperbolic identities
1
sinh it = sin t
i
cosh it = cos t,
∗ hLaplaceTransformOfCosineAndSinei created: h2013-03-2i by: hpahioi version: h40929i
Privacy setting: h1i hDerivationi h44A10i
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which enable the transition from hyperbolic to trigonometric functions. If we
choose α := ia in (3), we may calculate
cos at = cosh iat x
s
s
= 2
,
s2 −(ia)2
s + a2
the formula (4) analogously gives
sin at =
1
a
1
ia
= 2
.
sinh iat x · 2
2
i
i s −(ia)
s + a2
Accordingly, we have derived the Laplace transforms
L{cos at} =
s
,
s2 +a2
(5)
L{sin at} =
a
,
s2 +a2
(6)
which are true for s > 0.
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