The $c_0$ term in a Laurent expansion

Question

Is the $c_0$ term part of the principal part of a Laurent series, analytic part (positive powers in $z$) or neither or both?

I am working on a problem, where I have to match up some coefficients, but am not sure what to do with the $c_0$ term.

Answer 1

Given a function $$f(z) = \sum_{n=-\infty}^\infty c_n z^n,$$ holomorphic in a neighborhood of (but not at!) $0$, as you say, we can write it as the sum of the analytic part and a completely non-analytic part. The key point is that $\sum_{n=0}^\infty c_n z^n$ is analytic at $0$, whereas not a single term in $\sum_{n=-\infty}^{-1} c_n z^n$ is analytic. So this is the preferred way to split the sum, as far as I know.

If you have a more specific question about the problem you're looking at, I'm glad to answer it.