15.7 SOME THEOREMS ON LAPLACE TRANSFORMS
The property of linearity of Laplace transforms was already noted and made use of in earlier sections. We look at other interesting properties of Laplace transform in this section.
15.7.1 Time-shifting Theorem
If v(t) = f(t) u(t) has a Laplace transform V(s) then vd(t) = v(t – td) = f(t – td) u(t – td) has a Laplace transform Vd(s) = V(s)e–std.
The shifting operation implied in this theorem is illustrated in Fig. 15.7-1. Note that there is a difference between f(t – td) u(t) and f(t – td) u(t) u(t – td). Time-shifting theorem for unilateral Laplace transform works properly for f(t – td) u(t – td) but not for f(t – td) u(t).
Fig. 15.7-1 Illustrating the Time-Shift Operation Envisaged in Shifting ...