Engineering Tables/Laplace Transform Properties

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Property Definition
Linearity \mathcal{L}\left\{a f(t) + b g(t) \right\}  = a F(s)  + b G(s)
Differentiation \mathcal{L}\{f'\}  = s \mathcal{L}\{f\} - f(0^-)

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

Frequency Division \mathcal{L}\{ t f(t)\}  = -F'(s)

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

Frequency Integration \mathcal{L}\left\{ \frac{f(t)}{t} \right\} = \int_s^\infty F(\sigma)\, d\sigma
Time Integration \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  \mathcal{L} \left\{ f(at) \right\} = {1 \over a} F \left ( {s \over a} \right )
Initial value theorem f(0^+)=\lim_{s\to \infty}{sF(s)}
Final value theorem f(\infty)=\lim_{s\to 0}{sF(s)}
Frequency Shifts \mathcal{L}\left\{ e^{at} f(t) \right\}  = F(s - a)

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

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

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

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

Where:

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