Analytic iff Holomorphic on open domains of $\mathbb{C}$ I am reading through the proof of Griffiths and Harris on pages $2/3$. I am struggling with the proof of
$f$ is holomorphic iff $f$ is analytic.
They have a proof of holomorphicity and analaticity of function defined over open subsets of the complex plane. First I suppose that
$$\frac{\partial f}{\partial \overline{z}}=0.$$
I'd like to know where he uses this in proving there exists a converging power series, so first assuming $\frac{\partial f}{\partial \overline{z}}=0,$ does he use Cauchy Integral Formula? and this fact?
I also want to know what the most formal definition of $\textit{holomorphic}$ is vs. $\textit{analytic}$. From my understanding $f$ is holomorphic on some open $U \subset \mathbb{C}$ if for each $z \in U$, one has
$$\frac{\partial f}{\partial \overline{z}}=0.$$
And $f$ is analytic on all of $U$ is for each $z_0 \in U$ one can write $f$ as a power series centered at some $z_0 \in U$,
$$f(z)=\sum_{n=0}^\infty a_n(z-z_0)^n$$
such that the series converges on $U$.
First off, are these the more standard definitions of holo and analytic? Secondly, in proving holomorphic $\Rightarrow$ analytic, where do they use that the partial of the conjugate is zero? If anyone has a copy of Griffiths $\&$ Harris. Sorry if my question is too elementary, I am applying for PhDs for Fall 2022 and just wanna head start on some of the concepts I may encounter once there.
 A: The definition that an infinitely differentiable function $f$ is holomorphic on an open subset $U\subset\mathbb{C}$ if $$\frac{\partial f}{\partial \bar z}= 0 $$
a perfectly good definition and is essentially just a restatement of the Cauchy Riemann equations. It seems in Griffiths and Harris they prove a generalized version of the Cauchy integral formula which holds for any $C^{\infty}$ function (which reduces to the standard version when $\frac{\partial f}{\partial \bar z}=0$) and use this to prove that such functions are analytic.
If you want to learn about the equivalent definitions of holomorphic/complex analytic functions and theorems about their equivalence I'd suggest a book specifically on complex analysis.
A: I do not have a copy of that text book at hande, but the standard definition of holomorphic function is that it is a differentiable function, that is, that, for each $z_0$ in its domain, the limit
$$\lim_{z \rightarrow z_0}\frac{f(z)-f(z_0)}{z-z_0}$$
exists.Quite often, but not always, it is added to the definition the condition that the domain of $f$ is an open subset of $\Bbb C$.
