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3.3 Shifting and the Heaviside Function 89

15 15

10 10
5 5

–10 –5 0 5 10 15 20 –10 –5 0 5 10 15 20
t
–5 –5
t
–10 –10
–15 –15

FIGURE 3.13 t sin(t). FIGURE 3.14 H(t − 3/2)t sin(t).

10

5

0 –5 10 15 20
t
–5
–10

FIGURE 3.15 H(t − 4)(t − 4)
sin(t − 4).

THEOREM 3.4 Second Shifting Theorem

L[H(t − a) f (t − a)](s) = e −as F(s). (3.6)

This result follows directly from the definition of the transform and of the Heaviside
function.

EXAMPLE 3.8
Suppose we want L[H(t − a)]. Write
H(t − a) = H(t − a) f (t − a)

with f (t) = 1 for all t. Since F(s) = L[1](s) = 1/s, then by the second shifting theorem,
1
L[H(t − a)](s) = e −as F(s) = e −as .
s

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October 14, 2010 14:14 THM/NEIL Page-89 27410_03_ch03_p77-120