# Engineering Tables/Laplace Transform Properties

Property Definition
Linearity ${\displaystyle {\mathcal {L}}\left\{af(t)+bg(t)\right\}=aF(s)+bG(s)}$
Differentiation ${\displaystyle {\mathcal {L}}\{f'\}=s{\mathcal {L}}\{f\}-f(0^{-})}$

${\displaystyle {\mathcal {L}}\{f''\}=s^{2}{\mathcal {L}}\{f\}-sf(0^{-})-f'(0^{-})}$
${\displaystyle {\mathcal {L}}\left\{f^{(n)}\right\}=s^{n}{\mathcal {L}}\{f\}-s^{n-1}f(0^{-})-\cdots -f^{(n-1)}(0^{-})}$

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

${\displaystyle {\mathcal {L}}\{t^{n}f(t)\}=(-1)^{n}F^{(n)}(s)}$

Frequency Integration ${\displaystyle {\mathcal {L}}\left\{{\frac {f(t)}{t}}\right\}=\int _{s}^{\infty }F(\sigma )\,d\sigma }$
Time Integration ${\displaystyle {\mathcal {L}}\left\{\int _{0}^{t}f(\tau )\,d\tau \right\}={\mathcal {L}}\left\{u(t)*f(t)\right\}={1 \over s}F(s)}$
Scaling ${\displaystyle {\mathcal {L}}\left\{f(at)\right\}={1 \over a}F\left({s \over a}\right)}$
Initial value theorem ${\displaystyle f(0^{+})=\lim _{s\to \infty }{sF(s)}}$
Final value theorem ${\displaystyle f(\infty )=\lim _{s\to 0}{sF(s)}}$
Frequency Shifts ${\displaystyle {\mathcal {L}}\left\{e^{at}f(t)\right\}=F(s-a)}$

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

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

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

Convolution Theorem ${\displaystyle {\mathcal {L}}\{f(t)*g(t)\}=F(s)G(s)}$

Where:

${\displaystyle f(t)={\mathcal {L}}^{-1}\{F(s)\}}$
${\displaystyle g(t)={\mathcal {L}}^{-1}\{G(s)\}}$
${\displaystyle s=\sigma +j\omega }$