So, I've been studying Laurent series, and I'm fine with series such as $ \frac {1}{(z-1)(z+1)} $ for example. For these, we can just use partial fraction decomposition and then geometric series. However, I'm not even sure how to get started with the following function:
$ f(z)= (z^2+4)^\frac {1}{3}$
Obviously, I can see that there are zero's at $ +/- 2i $, but I have no idea how to even start to arrange this as a Laurent series. Do I do the same thing I would do for Taylor series? Any guidance would be appreciated.
Write $f(z) = \sqrt[3]{4} (({z \over 2})^2+1)^{1 \over 3}$, and use the binomial theorem to expand, this gives the Laurent series for $|z| < 2$.