Taylor serie convergence and holomorphic functions

Question

I have a doubt related to holomorphic functions and the convergence of their Taylor series. Is it true that:

"if and only if $f(z)$ is holomorphic in D (open), then Taylor expansion of $f(z)$ in D converges to $f(z)$ for all $z \in D$"

In other words, is it possible to have a function with poles in D, where its regular Taylor converges to $f(z)$ for some $z \in D$ but not necessarily for all?

A strong reference would be appreciated.

Answer 1

The statement

$f:D\to \mathbb C$ is holomorphic if and only if the Taylor expansion of $f$ converges to $f(z)$ for every $z\in D$.

is true.

For the $\implies$ direction, you can use the Cauchy integral formula.

For the $\impliedby$ direction I don't know any relevant reference, but it seems quite straightforward that if you assume that the Taylor series of the function converges for every $z$, then the function is at least differentiable (hence holomorphic) for every $z$.

The central part of the issue is the first statement, which is nicely proven in Theorem 10.16 of the third edition of Rudin's Real and Complex Analysis.

You should check this question to understand why the concept of analytic and holomorphic functions were born even though they ultimately identify the same functions over $\mathbb C$.