L { f ″ } = s 2 L { f } − s f ( 0 − ) − f ′ ( 0 − ) {\displaystyle {\mathcal {L}}\{f''\}=s^{2}{\mathcal {L}}\{f\}-sf(0^{-})-f'(0^{-})} L { f ( n ) } = s n L { f } − s n − 1 f ( 0 − ) − ⋯ − f ( n − 1 ) ( 0 − ) {\displaystyle {\mathcal {L}}\left\{f^{(n)}\right\}=s^{n}{\mathcal {L}}\{f\}-s^{n-1}f(0^{-})-\cdots -f^{(n-1)}(0^{-})}
L { t n f ( t ) } = ( − 1 ) n F ( n ) ( s ) {\displaystyle {\mathcal {L}}\{t^{n}f(t)\}=(-1)^{n}F^{(n)}(s)}
L − 1 { F ( s − a ) } = e a t f ( t ) {\displaystyle {\mathcal {L}}^{-1}\left\{F(s-a)\right\}=e^{at}f(t)}
L − 1 { e − a s F ( s ) } = f ( t − a ) u ( t − a ) {\displaystyle {\mathcal {L}}^{-1}\left\{e^{-as}F(s)\right\}=f(t-a)u(t-a)}
Where: