Analytic/Holomorphic Complex Functions

Recall from the Complex Differentiable Functions page that if $A \subseteq \mathbb{C}$ is open, $z_0 \in A$, and $f : A \to \mathbb{C}$, then $f$ is said to be complex differentiable at $z_0$ if the following limit exists:

(1)

\begin{align} \quad f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0} \end{align}

If this limit does exist, then the limit $f'(z_0)$ is called the derivative of $f$ at $z_0$.

We will now touch upon a new concept

Definition: Let $A \subseteq \mathbb{C}$ be open, $z_0 \in A$, and $f : A \to \mathbb{C}$. Then $f$ is said to be Analytic at $z_0$ or Holomorphic at $z_0$ if $f$ is complex differentiable on a neighbourhood containing $z_0$, i.e., there exists an $r > 0$ such that $D(z_0, r) \subseteq A$ and $f$ is complex differentiable on $D(z_0, r)$.

Definition: A function $f : \mathbb{C} \to \mathbb{C}$ is said to be an Entire Function if $f$ is analytic/holomorphic on all of $\mathbb{C}$.

Example 1

For a simple example, recall that if $f(z) = z$ then $f'(z_0) = 1$ for all $z_0 \in \mathbb{C}$. Since the complex derivative of $f$ exists at every point $z_0 \in \mathbb{C}$ it is clear that $f$ is analytic on $\mathbb{C}$ and of course, is analytic on every open subset of $\mathbb{C}$ and in particular, on all of $\mathbb{C}$. So $f(z) = z$ is actually an entire function.