Holomorphic function

Definition

Definition for one-variable function

Let $U$ be an open subset (not necessarily connected, though we can for practical purposes restrict ourselves to domains -- connected open subsets) of $\mathbb {C}$. A function $f:U\to \mathbb {C}$ is termed a holomorphic function if it satisfies the following equivalent conditions:

1. $f$ is complex-differentiable at every point of $U$
2. $f$ is complex-differentiable at every point of $U$, and the function $f':U\to \mathbb {C}$ we obtain as the derivative, is a continuous function.
3. $f$ is infinitely often complex-differentiable. In other words, we can take the $n^{th}$ derivative of $f$ for any $n$
4. For any point $z_{0}\in U$, and any disc centered at $z_{0}$ of radius $r$, that lies completely inside $U$, $f$ can be expressed using a power series in $(z-z_{0})$

Definition for functions in several variables

Let $U$ be an open subset (not necessarily connected, though we may restrict attention to connected subsets). A function $f:U\to \mathbb {C}$ is termed a holomorphic function if it satisfies the following equivalent conditions:

1. $f$ is complex-differentiable at every point of $U$
2. $f$ is complex-differentiable, and its partial derivatives in all directions are continuous functions
3. $f$ is infinitely often complex-differentiable. In other words, we can take any sequence of mixed partial derivatives of $f$
4. For any point $z_{0}\in U$, and any disc centered at $z_{0}$ of radius $r$, that lies completely inside $U$, $f$ can be expressed using a power series in $z-z_{0}$

Equivalence of definitions

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