TITLE: Meromorphic, analytic, holomorphic and all that
QUESTION [9 upvotes]: I must have slept through something in my complex variables course, because all my life I have used the terms holomorphic, meromorphic, and analytic somewhat interchangeably.  These are all also related to regular functions.
I have also thought of  "entire" and "everywhere analytic" as interchangeable terminology.
What are the distinctions between these terms?  And what is the correct terminology for a function which may have poles but not essential singularities.
(For example, $$e^{-\frac{1}{z^2}}$$ is in some sense nastier at $z=0$ than $z^{-4}$)?

REPLY [5 votes]: Let $\Omega \subset \mathbb{C}$ be open set.
A function $f : \Omega \to \mathbb{C}$ is called holomorphic if it is complex differentiable in any $z \in \Omega$. A holomorphic function $f : \mathbb{C} \to \mathbb{C}$ is called entire.
A function $f : \Omega \to \mathbb{C}$ is called analytic if it can be represented as a convergent power series in a neighborhood of each point $z \in \Omega$.
A function $f : \Omega \to \mathbb{C}$ is called meromorphic if it is holomorphic on $\Omega$ except for a set of poles, i.e., $f : \Omega \setminus P \to \mathbb{C}$ is holomorphic, where $P$ denotes the set of poles of $f$.