Engineering Tables/Laplace Transform Properties

Property	Definition
Linearity	$ℒ\{af(t)+bg(t)\}=aF(s)+bG(s)$
Differentiation	$ℒ\{f^{\prime }\}=sℒ\{f\}-f(0^{-})$ $ℒ\{f^{″}\}=s^2ℒ\{f\}-sf(0^{-})-f^{\prime }(0^{-})$ $ℒ\left\{f^{(n)}\right\}=s^nℒ\{f\}-s^{n-1}f(0^{-})-\cdots -f^{(n-1)}(0^{-})$
Frequency Division	$ℒ\{tf(t)\}=-F^{\prime }(s)$ $ℒ\{t^{n}f(t)\}=(-1{)}^n{F}^{(n)}(s)$
Frequency Integration	$ℒ\left\{\frac{f(t)}{t}\right\}=\underset{s}{\overset{\mathrm{\infty }}{\int }}F(\sigma )d\sigma$
Time Integration	$ℒ\{\underset{0}{\overset{t}{\int }}f(\tau )d\tau \}=ℒ\{u(t)*f(t)\}=\frac{1}{s}F(s)$
Scaling	$ℒ\{f(at)\}=\frac{1}{a}F\left(\frac{s}{a}\right)$
Initial value theorem	$f(0^{+})=\underset{s\to \mathrm{\infty }}{\lim }sF(s)$
Final value theorem	$f(\mathrm{\infty })=\underset{s\to 0}{\lim }sF(s)$
Frequency Shifts	$ℒ\{e^{at}f(t)\}=F(s-a)$ ${ℒ}^{-1}\{F(s-a)\}=e^{at}f(t)$
Time Shifts	$ℒ\{f(t-a)u(t-a)\}=e^{-as}F(s)$ ${ℒ}^{-1}\{e^{-as}F(s)\}=f(t-a)u(t-a)$
Convolution Theorem	$ℒ\{f(t)*g(t)\}=F(s)G(s)$

Where:

$f(t)={ℒ}^{-1}\{F(s)\}$

$g(t)={ℒ}^{-1}\{G(s)\}$

$s=\sigma +j\omega$