Given a function $f(z)$ with a Laurent series at $z=0$, is there a way to decompose it as $$f(z) = f^-(z) + a_0 + f^+(z)$$ where $$f^-(z) = \sum_{k<0} a_k z^k$$ and $$ f^+(z) = \sum_{k>0} a_k z^k,$$ if $f(z)$ has an essential singularity at that point? If not in general, are there some special cases where it's doable? Of course, I can always find the series, separate the sum and sum it out, but can it be done some other way, perhaps using some integral methods?
Here's why I want to do it: I have a particular generating function which contains both negative and positive powers, but I'm trying to "remove" the negative powers from it.
If $f$ is analytic for $0<|z|<r$ then for $0<a<|z|<b<r$ $$f(z)= \frac1{2i\pi} \int_{|s|=b}f(s)( \frac{1}{s-z}-\frac1s)ds+\frac1{2i\pi} \int_{|s|=b} \frac{f(s)}{s}ds-\frac1{2i\pi} \int_{|s|=a} \frac{f(s)}{s-z}ds$$
Expanding $1/(s-z)$ in power series in $s/z$ or $z/s$ this is how we prove that $f$ has a Laurent series.