Transformada de Laplace

Propiedades

Linealidad

${\displaystyle {\mathcal {L}}\left\{af(t)+bg(t)\right\}=a{\mathcal {L}}\left\{f(t)\right\}+b{\mathcal {L}}\left\{g(t)\right\}}$

Derivación

${\displaystyle {\mathcal {L}}\{f'(t)\}=s{\mathcal {L}}\{f(t)\}-f(0)}$

${\displaystyle {\mathcal {L}}\{f''(t)\}=s^{2}{\mathcal {L}}\{f(t)\}-sf(0)-f'(0)}$

${\displaystyle {\mathcal {L}}\left\{f^{(n)}(t)\right\}=s^{n}{\mathcal {L}}\{f(t)\}-s^{n-1}f(0)-\cdots -f^{(n-1)}(0)}$ =

=${\displaystyle {\mathcal {L}}\left\{f^{(n)}(t)\right\}=s^{n}{\mathcal {L}}\{f(t)\}-\sum _{i=1}^{n}s^{n-i}f^{(i-1)}(0)}$

Integración

${\displaystyle {\mathcal {L}}\left\{\int _{0}^{t}f(\tau )\,d\tau \right\}={1 \over s}{\mathcal {L}}\{f\}}$

Dualidad

${\displaystyle {\mathcal {L}}\{tf(t)\}=-F'(s)}$

Desplazamiento de la frecuencia

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

Desplazamiento temporal

${\displaystyle {\mathcal {L}}\left\{f(t-a)u(t-a)\right\}=e^{-as}F(s)}$

${\displaystyle {\mathcal {L}}^{-1}\left\{e^{-as}F(s)\right\}=f(t-a)u(t-a)}$