VIl2 Use the definition of the Laplace Transform to show tha

VIl.2. Use the definition of the Laplace Transform to show that (FIRST TRANSLATION THEOREM): le o) - Fis- a) Hence: 9 (F(s- a) (complete the statement)

Solution

Proof of : L{e^at f(t)} = F(s a) where F(s) = L{ f(t) }.

by definition, L{e^at f(t)} =   ₀ e^st e^at f(t) dt = ₀ e^(sa)t f(t)dt = F(s a)    (by definition)

This implies L¹{ F(s a) } = e^at f(t) .