I'm working on this inverse Laplace Transform problem
$$ \mathscr{L^{-1}}\left(\frac{(BI_{0}(\surd{s}) - \frac{CI_{1}(\surd{s})}{\surd{s}}}{(I_{0}(\surd{s}) - \frac{AI_{1}(\surd{s})}{\surd{s}}}\right) $$
where $I_{0}(x),I_{1}(x)$ are the modified Bessel Functions of the first kind, and $A,B$ and $C$ are constants. This is a transfer function for my system, which I'd like to have in the time domain to convolve my input with.
In [1], they derive a transfer function with the same form (different constants) and say this
The inverse Laplace transforms of the equation may be obtained easily by standard methods. We note that the equation is of the fractional form $\frac{p(s)}{q(s)}$, the inverse of which is $$\mathscr{L^{-1}}\left(\frac{p(s)}{q(s)}\right) = \sum_{n=1}^{\infty} \frac{p(\alpha_{n})}{q'(\alpha_{n})}e^{\alpha_{n}t} $$ where $p(s)$ and $q(s)$ are analytic at $s=\alpha_{n}$, $p(\alpha_{n}) \neq 0$, and $\alpha_{n}$ are simple poles of $\frac{p}{q}$
So, I have two questions I guess.
1) Where did this formula come from for the inverse? I've been searching but can't find any other sources for this. I assume this comes from some sort of partial fraction decomposition into an infinite series?
2) Can this inverse even be found directly? Residue theorem doesn't seem applicable, as Jordan's lemma is not satisfied (The function does not tend to 0 as the radius of the contour approaches infinity, it tends towards some constant) Partial fraction decomposition seems plausible, but I'm having trouble manipulating the series representations of the modified Besesel functions. Also, the powers of the numerator and denominator are equal. Would I need to find the inverse of the transfer function convolved with the step function $\frac{1}{s}$, and then take the derivative in the time domain?
Any help or hints you guys could provide would be greatly appreciated.
[1] Mow, V. C. (1984). An analysis of the unconfined compression of articular cartilage. Journal of biomechanical engineering, 106, 165.
Edit: I've mostly solved my problem. Here's some information on each of these I was having trouble with if anyone else sees this.
1) This is called the Heaviside decomposition theorem.
2) A good chapter detailing this inverse can be found in chapter 7 Biomechanical Aspects of Soft Tissues