Table of Laplace transforms

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Section 3.8 Table of Laplace transforms

Table 3.8.1. Laplace transform table

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$f(t)=$	$\mathcal{L}\{f\}(s)\equiv F(s)=$
$1$	$\dfrac{1}{s}$
$e^{at}$	$\dfrac{1}{s-a}$
$t^n$	$\dfrac{n!}{s^{n+1}}, n=1,2,3,\ldots$
$\sin(kt)$	$\dfrac{k}{s^2+k^2}$
$\cos(kt)$	$\dfrac{s}{s^2+k^2}$
$f'(t)$	$sF(s)-f(0)$
$f''(t)$	$s^2F(s)-sf(0)-f'(0)$
$e^{at}f(t)$	$F(s-a)$
$\mathcal{U}(t-a)$	$\dfrac{e^{-as}}{s}$
$f(t-a)\mathcal{U}(t-a)$	$e^{-as}F(s)$
$\delta(t-a)$	$e^{-as}$
$(f*g)(t)$	$F(s)G(s)$

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\(f(t)=\)	\(\mathcal{L}\{f\}(s)\equiv F(s)=\)
\(1\)	\(\dfrac{1}{s}\)
\(e^{at}\)	\(\dfrac{1}{s-a}\)
\(t^n\)	\(\dfrac{n!}{s^{n+1}}, n=1,2,3,\ldots\)
\(\sin(kt)\)	\(\dfrac{k}{s^2+k^2}\)
\(\cos(kt)\)	\(\dfrac{s}{s^2+k^2}\)
\(f'(t)\)	\(sF(s)-f(0)\)
\(f''(t)\)	\(s^2F(s)-sf(0)-f'(0)\)
\(e^{at}f(t)\)	\(F(s-a)\)
\(\mathcal{U}(t-a)\)	\(\dfrac{e^{-as}}{s}\)
\(f(t-a)\mathcal{U}(t-a)\)	\(e^{-as}F(s)\)
\(\delta(t-a)\)	\(e^{-as}\)
\((f*g)(t)\)	\(F(s)G(s)\)