# Meromorphic function at a point.

i'm study meromorphic function at the complex plane extended, but i have a trouble. I know if $$f \colon D \subseteq \mathbb{C} \to \mathbb{C}$$ $$f$$ is meromorphic on D if the singularities of $$f$$ are poles in $$D$$ But whats mean $$f$$ is meromorphic at a point $$a \in \mathbb {C}$$?

Other quickly question, anyone knows why $$\infty$$ is branched point of $$log(z)$$?

Thanks

• In answer to your second question: $\infty$ is a branch point of $\log z$ because $0$ is a branch point of $\log \frac{1}{z} = -\log z$. – mjw Jun 12 '19 at 4:47
• We usually don't define holomorphicity, meromorphicity, etc. at single points because singletons are closed and we need open sets for almost all the nice theorems. I would thus assume that when someone writes that "$f$ is meromorphic at $a$" the author is saying "$f$ is meromorphic in a neighborhood of $a$". However, without knowing the context you found this it is hard to say – Brevan Ellefsen Jun 12 '19 at 5:14
• I've found this in "Complex functions: An algebraic and geometric viewpoint" - Singerman. The point of this is extend the notions of analytic and meromorphic functions at complex plane extend, autor says $f(z)$ are analytic, meromorphic, etc... at infinity if $f(1/z)$ are analytic, meromorphic, etc... At $0$. – Davis We Jun 12 '19 at 12:31
• @DavisWe Oh, then yes - the author means analytic in a neighborhood. This is very standard terminology. You are essentially looking at some small neighborhood of the north pole on the Riemann sphere and seeing if you get holo/mero-morphicity. (For some sort of reference, here's a post on this site from a quick google search. Let me know if you still have questions on this) – Brevan Ellefsen Jun 13 '19 at 5:14