A question about the definition of analyticity I am reading Gamelin's textbook on Complex Analysis.
In this book, he defines a function is analytic at an open set $U$ if the function is complex differentiable at every point in $U$ and the the derivative is continuous on $U$.
But what would be the definition of a function that is $\textbf{analytic at a point $z_0$}$?
I also know that some authors use the term holomorphic, what's the difference between those?
 A: Originally, there was a big difference between analytic and holomorphic functions. Their definition was:
Let $U\subseteq\mathbb C$ open, $f:U\to\mathbb C$. Then $f$ is called conplex differentiable in $z_0\in U$ if $\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}$ exists. It is called holomorphic in an open subset $V\subseteq U$ if it is complex differentiable in all $z\in V$, and just holomorphic if it is holomorphic in $U$. It is called analytic in $z_0$ if there exists an open neighborhood of $z_0$ in which it agrees with a power series centered at $z_0$. And it is called analytic in an open subset $V\subseteq U$ if it is analytic in all $z\in V$, and just analytic if it is analytic in $U$.
Now there are some non-obvious implications. For one, if $f$ is analytic in a point, it is analytic in an open neighborhood of that point. This might seem obvious at first, because analytic means that it agrees with a power series on an open neighborhood by definition. But the power series has to be centered at that point. It is not obvious that there exists a power series $\sum_k b_k(z-z_2)^k$ with which $f$ agrees, just because $f$ agrees with a power series $\sum_k a_k(z-z_1)^k$ on a set encompassing $z_2$. But as it turns out, analytic in a point implies analytic in an open neighborhood. So it's sufficient to define analyticity for open sets.
The bigger, even less obvious implication is that analyticity and holomorphicity imply each other: a function is analytic in an open set if and only if it is holomorphic in that same set. This is the starting point for an entire host of very strong theorems about holomorphic functions, so I think it's worth pointing out that analytic and holomorphic are originally distinct, but then turn out to be the same thing.
