How to find $$\mathcal{L^{-1}} \left[ \frac{F(s)}{s+a} \right]$$where $F(s)$ is the Laplace transform of $f(t).$
How to find the inverse laplace transform of an arbitrary function
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2The title is seriously misleading. – 2012-10-15
3 Answers
Use laplace transform properties : $$ G(s)=\frac{F(s)}{s+a} $$ $$ \mathcal{L^{-1}}[G(s)]=g(t) $$ $$ \mathcal{L} \left[ e^{at}g(t) \right] = G(s-a)=\frac{F(s-a)}{s} $$ $$ \frac{1}{s} \triangleq \int_0^t $$ $$ e^{at}g(t)=\int_0^t e^{a \tau }f( \tau ) d \tau $$ $$ g(t)=e^{-at} \int_0^t e^{a \tau }f( \tau ) d \tau $$ or we can write : $$ g(t)= \int_0^t e^{-a(t- \tau) }f( \tau ) d \tau $$
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0Nice approach +1. – 2012-12-22
Hint:
You know that $$\mathcal{L}(f*g)=F(s)G(s)$$ so $$\mathcal{L^{-1}}\big(F(s)G(s)\big)=f*g$$ wherein $f*g=\int_0^tf(\kappa)g(t-\kappa)d\kappa$.
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0Great hint! + 1 – 2013-02-12
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0@Thanks. I hope I do well for to day at M.S.E. – 2013-02-12
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0You're doing well for MSE! – 2013-02-12
If you know about convolution, this is just a piece of cake.
$\mathcal L^{-1}\left\{\dfrac{F(s)}{s+a}\right\}$
$=\mathcal L^{-1}\left\{\dfrac{1}{s+a}\right\}*\mathcal L^{-1}\{F(s)\}$
$=e^{-at}*f(t)$
$=\int_0^te^{-a(t-\tau)}f(\tau)~d\tau$
$=e^{-at}\int_0^te^{a\tau}f(\tau)~d\tau$