First Shifting Property | Laplace Transform

First Shifting Property
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   when   $s > a$   then,

$\mathcal{L} \left\{ e^{at} \, f(t) \right\} = F(s - a)$


In words, the substitution   $s - a$   for   $s$   in the transform corresponds to the multiplication of the original function by   $e^{at}$.

Proof of First Shifting Property
$\displaystyle F(s) = \int_0^\infty e^{-st} f(t) \, dt$

$\displaystyle F(s - a) = \int_0^\infty e^{-(s - a)t} f(t) \, dt$

$\displaystyle F(s - a) = \int_0^\infty e^{-st + at} f(t) \, dt$

$\displaystyle F(s - a) = \int_0^\infty e^{-st} e^{at} f(t) \, dt$

$F(s - a) = \mathcal{L} \left\{ e^{at} f(t) \right\}$       okay



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