Meromorphic, analytic, holomorphic and all that

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 $f$.

Holomorphic means complex differentiable on some open set. Analytic means has a power series expansion on some open set. A remarkable result of complex analysis is that these are equivalent.

Meromorphic means holomorphic except at isolated points which are specifically poles. Thus $z^{-4}$ is meromorphic while $e^{-1/z^2}$ is not.