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any function f.
. . . . .

2. Properties of Laplace Transform

Linearity

Theorem 2.1È

Let f(t) and g(t) be functions whose Laplace transforms exist and let α
and β be scalars. Then

L[αf(t) + βg(t)] = αL[f(t)] + βL[g(t)].
. . . . . . .

PROOF. Using the definition of the Laplace transform

∞ ∞ ∞
∫ ∫ ∫
e −st (αf(t) + βg(t)) dt = α e −st f(t) dt + β e −st g(t) dt.

0 0 0
. . . .
.

Example 2.1, f(t) = cos bt. Evaluate Lf(t).
. . . . .
.

Solution. To compute the Laplace transform we will use the Euler formula
and Property of Linearity

iθ
e = cos θ + i sin θ,

which implies that
iθ
e + e −iθ
cos θ = .
2
2
2
2
Therefore, using i = −1 and (s − ib)(s + ib) = s + b we can write:
[ ] ( )
e ibt + e −ibt 1 1 1 s
L[cos bt] = L = + = .
2
2 2 s − ib s + ib s + b 2
Thus,
s
L[cos bt] = .
2
s + b 2
. . . .

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