I tried to find this in Churchill's Operational Mathematics which has a good variety of transform pairs, but no matches for what appears a simple expression. Does anyone have a solution for the inverse Laplace Transform of
$$\sqrt{s+a\over s+b}$$
where a and b are both real valued constants?
$\mathcal{L}^{-1}_{s\to t}\sqrt{\dfrac{s+a}{s+b}}$
$=e^{-bt}\mathcal{L}^{-1}_{s\to t}\sqrt{\dfrac{s+a-b}{s}}$
$=e^{-bt}\left(\dfrac{(a-b)e^\frac{(b-a)t}{2}}{2}\left(I_1\left(\dfrac{(a-b)t}{2}\right)+I_0\left(\dfrac{(a-b)t}{2}\right)\right)+\delta(t)\right)$ (according to http://eqworld.ipmnet.ru/en/auxiliary/inttrans/LapInv3.pdf)
$=\dfrac{(a-b)e^{-\frac{(a+b)t}{2}}}{2}\left(I_1\left(\dfrac{(a-b)t}{2}\right)+I_0\left(\dfrac{(a-b)t}{2}\right)\right)+\delta(t)$