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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 $z^{-4}$)?

Mark Fischler
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  • In the complex plane, a complex function is holomorphic if-f it is analytic. Whereas, meromorphic on an open subset means holomorphic everywhere, except some finite number of isolated points called poles. –  May 26 '16 at 15:10

2 Answers2

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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 $f$.

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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.

Ian
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    Yes, I remember that "remarkable" theorem from the course. What I had missed is the fact that absent that insight, holomorphic and analytic are two distinct concepts, and I guess I should be using "analytic" when I will be dealing with series expansions and "holomorphic" when the important consideration is differentiability or the Cauchy conditions. – Mark Fischler May 26 '16 at 17:26
  • @MarkFischler Once you get used to that theorem, you tend to just use whichever term you like better. But if you're not yet really comfortable, then yes, you should use the term that more closely matches the tools you are actually using. – Ian May 26 '16 at 17:31