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Translation theorems
.

b) By definition
at
e + e −at
cosh (at) = ,
2
therefore
[ at −at]
e + e 1 at 1 −at
L = L[e ] + L[e ] =
2 2 2
1 1 1 1 1 s + a + s − a s
= · + · = · = .
2
2 s − a 2 s + a 2 (s − a)(s + a) s − a 2
c) Similarly
at
e − e −at
sinh (at) = ,
2
therefore
[ at −at]
e − e 1 1
at
L = L[e ] − L[e −at ] =
2 2 2
1 1 1 1 1 s + a − s + a a
= · − · = · = .
2
2 s − a 2 s + a 2 (s − a)(s + a) s − a 2
. . . .

Translation theorems

We present two very useful results for determining Laplace transforms and their
inverses. The first pertains to a translation in the s-domain and the second to a
translation in the t-domain.

Theorem 2.2È

(First Translation Theorem) (First Shifting Theorem). If F(s) = L(f(t))

for Re(s) > 0, then

at
F(s − a) = L(e f(t)), (a real, Re(s) > a).
. . . . . . .

PROOF. For Re(s) > a,

∫ ∫
∞ ∞
e f(t)dt = L(e f(t)).
F(s − a) = e −(s−a)t f(t)dt = e −st at at
0 0
. . . .
.

Example 2.6,
2
2 3t 2
L[t e ] = L[t ] = 3 .
s−3 (s − 3)
. . . . .

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