Analytic iff Holomorphic on open domainss of C

Step 1
The definition that an infinitely differentiable function f is holomorphic on an open subset ${\displaystyle U\subset \mathbb{C}}$ if
${\displaystyle \frac{\partial f}{\partial \bar{z}}=0}$
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 ${\displaystyle C^{\mathrm{\infty }}}$
function (which reduces to the standard version when $\frac{\partial f}{\partial \overline{z}}=0$) 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.

Step 1
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 ${\displaystyle z_0}$ in its domain, the limit
${\displaystyle \underset{z\to z_0}{lim}\frac{f\left(z\right)-f\left(z_0\right)}{z-z_0}}$
exists.Quite often, but not always, it is added to the definition the condition that the domain of f is an open subset of ${\displaystyle \mathbb{C}}$