Holomorphic/Analytic functions Let $f$ be a function. Then $f$ is holomorphic is equivalent to $f$ being analytic which in turn is also equivalent to satisfying the Cauchy-Riemann equations. All three concepts imply infinite differentiability. 
My question is what do we need to add to infinite differentiability to recover analyticity/ holomorphicity/Cauchy Riemann equations?
 A: There are a lot of equivalent characterizations of holomorphic functions, but the main part of the story is that for a given function
$$
f: \mathbb{R}^2 \mapsto \mathbb{R}^2
$$
all you need to know is that $f$ is continuous. Then some equivalent conditions are:


*

*$f$ is complex differentiable (the differential $d f$ exists and is complex linear),

*$f$ is locally integrable (edit: I meant to say that $f$ has an antiderivative, this point and point 3 are usually called Morera's theorem),

*integrals along closed paths of $f$ are zero,

*$f$ is conformal,

*Cauchy's integral formula holds for $f$,

*$f$ is represented by a power series around every point,

*the Cauchy-Riemann equations hold,
etc. 
For some of these conditions to make sense you need to know that $f$ is once differentiable as a real function. But none are related to $f$ being smooth (infinitely often differentiable as a real function), so knowing that $f$ is smooth is knowledge that is orthogonal to knowing that $f$ is holomorphic.
Edit: Of course every holomorphic function is smooth. What I meant to say is that being smooth is not "closer to being holomorphic than simply being differentiable" in any sense. AFAIK, there is no statement like "a continuous function is holomorphic if it is smooth + some other condition".
