Complex differentiability, holomorphicity, analyticity - what's the difference?

Question

From what I gather, a function $f:\mathbb{C} \to \mathbb{C}$ is:

-Differentiable at $z_0$ iff the limit $\lim_{z \to z_0} \frac{f(z)-f(z_0)}{z-z_0}$ exists.

-Holomorphic at $z_0$ iff there exists a neighbourhood of $z_0$ in which $f$ is differentiable.

-Analytic at $z_0$ iff there exists a neighbourhood of $z_0$ in which $f(z) = \sum_{n=0}^{\infty} a_n (z-z_0)^n$.

And it is a theorem of complex analysis that the last two are equivalent.

Is this more or less right? Please correct me if I've made any mistakes.

Yes, it's right. It may be better to speak explicitly of complex differentiability to distinguish it from real differentiability, but that's anybody's choice. — Daniel Fischer, Sep 24 '13 at 14:14

score 2 · Accepted Answer · answered Sep 24 '13 at 20:31

2

You are correct.

(Answer added to keep the question off the unanswered list.)

answered Sep 24 '13 at 20:31

mrf

1 Answers1