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I need to find the inverse Laplace transform with respect to $s$ of the following function:

$$\tilde{F}\left(s,y,\omega\right)=\dfrac{s\cos\left(py\right)\cos\left(qd\right)}{4s^{2}pq\sin\left(pd\right)\cos\left(qd\right)+\left(q^{2}+s^{2}\right)^{2}\cos\left(pd\right)\sin\left(qd\right)}$$

where:

$$p=\sqrt{\frac{\omega^{2}}{c_{L}^{2}}+s^{2}}\qquad q=\sqrt{\frac{\omega^{2}}{c_{T}^{2}}+s^{2}}$$

Here $c$ is a positive constant and $\omega$, $y$ can take real values.

This is quite a monster and I don't know if there is a closed form for $\mathcal{L}^{-1}\left[\tilde{F}\right]$. So my question is twofold. Can you tell whether $\mathcal{L}^{-1}\left[\tilde{F}\right]$ will have a closed form? If there is a closed form, which is it? Moreover, any suggestions to simplify are welcome. Thanks.

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  • $\begingroup$ I'm not too good with Latex. Can someone edit somehow the big equation to make it fit? $\endgroup$ – becko Nov 26 '11 at 20:46
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    $\begingroup$ Set, e.g., $\alpha=\omega^2+s^2$, etc.. (btw, if you typeset that, you are good at Latex :) ) $\endgroup$ – David Mitra Nov 26 '11 at 21:00
  • $\begingroup$ @DavidMitra: thanks $\endgroup$ – becko Nov 30 '11 at 3:54
  • $\begingroup$ Mathematica couldn't do it outright with `InverseLaplaceTransform'... $\endgroup$ – JohnD Dec 21 '12 at 23:12
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Liouville's "Rational Integral Theorem" may be of some use for your first question, although I'm not that sure without looking into it further. See Fitt's paper, p.230.

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