Electronics/Laplace Transform pairs

$f(t)$	$F(s)$
$\delta (t-t_{0})$	$e^{-st_{0}}$
$u(t)$	${\frac {1}{s}}$
$e^{-\alpha t}u(t)$	${\frac {1}{s+\alpha }}$
$\sin(\beta t)u(t)$	${\frac {\beta }{s^{2}+\beta ^{2}}}$
$\cos(\beta t)u(t)$	${\frac {s}{s^{2}+\beta ^{2}}}$
$e^{-\alpha t}\sin(\beta t)u(t)$	${\frac {\beta }{(s+\alpha )^{2}+\beta ^{2}}}$
$e^{-\alpha t}\cos(\beta t)u(t)$	${\frac {s+\alpha }{(s+\alpha )^{2}+\beta ^{2}}}$
$2e^{\alpha t}(a\cos {\beta t}-b\sin {\beta t})u(t)$	${\frac {a+jb}{s+\alpha -j\beta }}+{\frac {a-jb}{s+\alpha +j\beta }}$
$t^{n}$	${\frac {n!}{s^{n+1}}}$
$t^{n}f(t)$	$(-1)^{n}{\frac {d^{n}F(s)}{ds^{n}}}$
${\frac {df(t)}{dt}}$	$sF(s)-f(0)$
$\int _{0}^{t}f(x)dx$	${\frac {1}{s}}F(s)$
$u_{-1}(t-t_{0})f(t-t_{0})$	$e^{-t_{0}s}F(s),t_{0}\geq 0$
$h(t)*f(t)=\int _{0}^{t}h(t-\tau )f(\tau )d\tau$	$H(s)F(s)$
$f_{1}(t)f_{2}(t)$	$F_{1}(s)*F_{2}(s)$

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