Analytic iff Holomorphic on open domains of $\mathbb{C}$

Solution 1:

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.