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Monday, 25 March 2013

TABLE OF LAPLACE TRANSFROM


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)
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