Question: inverse LaPlace transform

Included below is vector partial diff eq I am working with.  To get rid of the time deriv's I took the LaPlace transform & the remaining spatial eq in the s-domain is listed.  To make matters simpler I set beta = 0 to get rid of the curl of the field.  What remains is essentially the Helmholtz eq.  To simplify further I just found the homogeneous soln for the x direction only.

As can be seen the eigenfunctions are exponentials with s beneath a square root sign.  I have not had any success at inverting back into the time domain.  I also tried the residue theorem without success.  Should I be employing a different transform or is there some trick to get the inversion to work with this solution form?

As far as I can tell I executed the forward transform correctly.  The dimensions beneath the sq root sign is 1/length^2 which then multiplied by the lenght dimension x gives dimensionless units for the exponential.  So I am reasonable confident there.

Suggestions???

Download inv_laplace_computat.mw