Engineering Tables/Laplace Transform Properties

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

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