Analytic Complex Functions

Analytic Complex Functions

Recall from the Differentiable Complex 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 $z_0$ if there exists an $r > 0$ such that $f$ is complex differentiable on the open disk $D(z_0, r)$. $f$ is said to be Analytic on $A$ if $f$ is analytic at each $z_0 \in A$.

The term "holomorphic" can be used interchangeably with "analytic" in this context.

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 $\mathbb{C}$.

We will now state a very important theorem which says that if $f$ is complex differentiable at a point $z_0$, i.e., if $f'(z_0)$ exists, then $f$ is necessarily continuous at $z_0$. This will be important in proving some results regarding analytic functions shortly.

Theorem 1: Let $A \subseteq \mathbb{C}$ be open, $z_0 \in A$, and $f : A \to \mathbb{C}$. If $f$ is complex differentiable at $z_0$ then $f$ is continuous at $z_0$.
  • Proof: Since $f$ is complex differentiable at $z_0$, $f'(z_0)$ exists and so:
(2)
\begin{align} \quad \lim_{z \to z_0} [f(z) - f(z_0)] &= \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0} \cdot (z - z_0) \\ &= f'(z_0) \cdot \lim_{z \to z_0} (z - z_0) \\ &= f'(z_0) \cdot 0 \\ &= 0 \end{align}
  • Therefore $\displaystyle{\lim_{z \to z_0} f(z) = f(z_0)}$ and so $f$ is continuous at $z_0$. $\blacksquare$
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