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I am trying to find the inverse Laplace transform $(g(t))$ of

$ G(s) = \frac{2s}{(s+1)^2+4}$

I know about the inverse transforms $e^{a t}\cos(\omega t)$ and $\mathrm{e}^{at}\sin(\omega t)$ however I am trying to get the inverse transforms without these, as these are not on the table of standard transforms for our course.

I was also attempting to use $\mathrm{e}^{at} f(t) \leftrightarrow F(s-a)$, but I'm not sure how to go about this, either.

Any help would be appreciated. :)

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    If you don't mind a little complex arithmetic then you can also derive both shifted sine and cosine from $\mathcal{L}^{-1}(\frac{1}{s-c}) = e^{ict}$ with $c=a+ib$. Just multiply by $\frac{s-(a-ib)}{s-(a-ib)}$ and see how the denominator goes to $(s-a)^2+b^2$ whereas the numerator has $(s-a)$ and $ib$. Since $e^{(a+ib)t} = e^{at}\cos(bt)+ie^{at}\sin(bt)$ equating real and imaginary parts reveals both the transform for sine and cosine at once.2012-09-02

2 Answers 2

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If you have $(s+1)^2$ in the denominator then make it appear in the numerator by adding zero:

$ G(s) = \frac{2s}{(s+1)^2+4} = \frac{2(s+1-1)}{(s+1)^2+4} = 2 \frac{s+1}{(s+1)^2+4}-\frac{2}{(s+1)^2+4} $

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The inverse Laplace transform is defined by,

$ f(x) = \frac{1}{2\pi i}\int_C F(s) {\rm e}^{sx} ds \,$

where $F(s)$ is the Laplace transform of $f(x)$, and $C$ is the Bromwich contour.

$ f(x) = \frac{1}{2\pi i} \int_C \frac{{2s\,\rm e}^{sx}}{((s+1)^2+4)} ds = \frac{1}{2\pi i} \int_C \frac{{2s\,\rm e}^{sx}}{(s^2+2s+5)} ds $

$= \frac{1}{2 \pi i} \int_C \frac{{2s\,\rm e}^{sx}}{(s+1+2i)(s+1-2i)} ds\,.$

The integrand has two simple poles at $s=-(1+2i)$ and $ s=-1+2i $ which implies,

$ f(x) = \lim_{s\rightarrow -(1+2i)} \frac{{2s\,\rm e}^{sx}}{(s+1-2i)} + \lim_{s\rightarrow (-1+2i)} \frac{{2s\,\rm e}^{sx}}{(s+1+2i)} $

$ = {{\rm e}^{-x}} \left( 2\,\cos \left( 2\,x \right) -\sin \left( 2\,x \right) \right) $