Propiedades Linealidad L { a f ( t ) + b g ( t ) } = a L { f ( t ) } + b L { g ( t ) } {\displaystyle {\mathcal {L}}\left\{af(t)+bg(t)\right\}=a{\mathcal {L}}\left\{f(t)\right\}+b{\mathcal {L}}\left\{g(t)\right\}} Derivación L { f ′ ( t ) } = s L { f ( t ) } − f ( 0 ) {\displaystyle {\mathcal {L}}\{f'(t)\}=s{\mathcal {L}}\{f(t)\}-f(0)} L { f ″ ( t ) } = s 2 L { f ( t ) } − s f ( 0 ) − f ′ ( 0 ) {\displaystyle {\mathcal {L}}\{f''(t)\}=s^{2}{\mathcal {L}}\{f(t)\}-sf(0)-f'(0)} L { f ( n ) ( t ) } = s n L { f ( t ) } − s n − 1 f ( 0 ) − ⋯ − f ( n − 1 ) ( 0 ) {\displaystyle {\mathcal {L}}\left\{f^{(n)}(t)\right\}=s^{n}{\mathcal {L}}\{f(t)\}-s^{n-1}f(0)-\cdots -f^{(n-1)}(0)} = = L { f ( n ) ( t ) } = s n L { f ( t ) } − ∑ i = 1 n s n − i f ( i − 1 ) ( 0 ) {\displaystyle {\mathcal {L}}\left\{f^{(n)}(t)\right\}=s^{n}{\mathcal {L}}\{f(t)\}-\sum _{i=1}^{n}s^{n-i}f^{(i-1)}(0)} Integración L { ∫ 0 t f ( τ ) d τ } = 1 s L { f } {\displaystyle {\mathcal {L}}\left\{\int _{0}^{t}f(\tau )\,d\tau \right\}={1 \over s}{\mathcal {L}}\{f\}} Dualidad L { t f ( t ) } = − F ′ ( s ) {\displaystyle {\mathcal {L}}\{tf(t)\}=-F'(s)} Desplazamiento de la frecuencia L { e a t f ( t ) } = F ( s − a ) {\displaystyle {\mathcal {L}}\left\{e^{at}f(t)\right\}=F(s-a)} Desplazamiento temporal L { f ( t − a ) u ( t − a ) } = e − a s F ( s ) {\displaystyle {\mathcal {L}}\left\{f(t-a)u(t-a)\right\}=e^{-as}F(s)} 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)} Nota: u ( t ) {\displaystyle u(t)} es la función escalón unitario.