Hyperphysics ajc: Laplace Transforms

terça-feira, 22 de abril de 2014

Elementary Laplace Transforms

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

Hyperphysics ajc