Graduate Inverse Laplace of an Overwhelming Function

[Floro Ortiz]

Hello, guys. I'm currently working on a physics problem that requires me to evaluate the inverse Laplace of the function in the attached file. When b = 0, "y" vanishes, and all one has to do is to look up the Laplace table for the inverse. However, non-zero b has been giving me a headache. I have already tried redefining variables, but my attempts haven't really simplified the problem. Any suggestions on how to attack this problem? Or, is the function integrable to begin with?

Thank you very much in advance. All inputs will be greatly appreciated.

 

[Santilopez10]

have you tried using complex analysis?

 

[Floro Ortiz]

have you tried using complex analysis?

Yes, but I always get stuck at the branch cut part of the integral. I tried decomposing the radicand to elminate the "smaller" square root, but the function only got harder to interpret.