Elementary Laplace Transforms

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Jun 3, 2019
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- Elementary Fourier Transforms
- Fundamental Trigonometric Identities

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$\mathcal{L} \{1\}(s)$	=	$\dfrac{1}{s} \; (s>0)$	(1)
$\mathcal{L} \{e^{at}\}(s)$	=	$\dfrac{1}{s-a} \; (s>a)$	(2)
$\mathcal{L} \{t^{n}\}(s)$	=	$\dfrac{n!}{s^{n+1}} \;\; (s>0, n \text{ is a positive integer})$	(3)
$\mathcal{L} \{t^{p}\}(s)$	=		4)
$\mathcal{L} \{\sin (at)\}(s)$	=	$\dfrac{a}{s^2+a^2} \; (s>0)$	(5)
$\mathcal{L} \{\cos (at)\}(s)$	=	$\dfrac{s}{s^2+a^2} \; (s>0)$	(6)
$\mathcal{L} \{e^{at} \cdot \sin (bt)\}(s)$	=	$\dfrac{b}{(s-a)^2+b^2} \; (s>a)$	(7)
$\mathcal{L} \{e^{at} \cdot \cos (bt)\}(s)$	=	$\dfrac{s-a}{(s-a)^2+b^2} \; (s>a)$	(8)
$\mathcal{L} \{t^{n} \cdot e^{at}\}(s)$	=	$\dfrac{n!}{(s-a)^{n+1}} \; (s>a)$	(9)
$\mathcal{L} \{t^{n} \cdot f(t)\}(s)$	=	$(-1)^n \dfrac{d^n}{ds^n} \mathcal{L} \{f(t)\}(s)$	(10)
$\mathcal{L} \{f'(t)\}(s)$	=	$s \cdot \mathcal{L} \{f(t) \} - f(0)$	(11)
$\mathcal{L} \{H_c(t)\}(s)$	=	$\dfrac{e^{-cs}}{s} \; (s>0)$	(12)
$\mathcal{L} \{H_c(t) \cdot f(t-c)\}(s)$	=	$e^{-cs} \mathcal{L}\{f(t)\}(s)$	(13)
$\mathcal{L} \{H_c(t) \cdot f(t)\}(s)$	=	$e^{-cs} \mathcal{L}\{f(t+c)\}(s)$	(14)
$\mathcal{L} \{ \delta_c(t)\}(s)$	=	$e^{-cs}$	(15)
$\mathcal{L} \{e({ct} cdot f(t)\}(s)$	=	$\mathcal{L} \{f(t)\}(s-c)$	(16)

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