Show that, provided a>0 and f is a real function that :
$L\left[ f\left( t-a\right) H\left( t-a\right) \right] =e^{-pa}L\left( f\left( t\right) \right)$
I understand that when we multiply a function $f(x)$ by $e^{px}$ we have :
$F\left( p-p_{0}\right)$
$$ \begin{align} \mathcal L\left[ f\left( t-a\right) H\left( t-a\right) \right] &=\int_a^\infty\mathrm e^{-pt}f\left( t-a\right)\mathrm dt\\ &=\int_0^\infty\mathrm e^{-p(u+a)}f\left( u\right)\mathrm du\qquad (t-a=u)\\ &=\mathrm e^{-pa}\int_0^\infty\mathrm e^{-pt}f\left( t\right)\mathrm dt\qquad (t=u)\\ &=\mathrm e^{-pa}\mathcal L\left[ f\left( t\right) \right] \end{align} $$