Holomorphic function

Definition

Let $Ω$ be an open subset of $C$ . A function $f : U \to C$ is termed a holomorphic function if it satisfies the following equivalent conditions:

$f$ is complex-differentiable at every point of $U$
$f$ is complex-differentiable at every point of $U$ , and the function $f^{'} : U \to C$ we obtain as the derivative, is a continuous function.
$f$ is infinitely often complex-differentiable. In other words, we can take the $n^{t h}$ derivative of $f$ for any $n$
For any point $z_{0} \in U$ , and any disc centered at $z$ of radius $r$ , that lies completely inside $U$ , $f$ can be expressed using a power series in $(z - z_{0})$

Definitions (1) and (2) are equivalent as a consequence of Morera's theorem.