Forgive me if this takes a while to post, attempting to get the LaTeX right.

Attempting to find a Laurent series expansion for the function:

Over 2 regions:

I have seen similar problems like:

where you can simplify by taking

Which is solved here

I can't seem to find a way to simplify my equation in a similar manner.

I know that (obviously):

Not sure where to go from there. If I knew a Maclaurin series for something like and or how to find one, that would make things much simpler as I could just multiply the series together...?

Thanks in advance to anyone who read all that.