📂Complex Anaylsis

Analytic Functions

Definition

Given open sets $A \subset \mathbb{C}$ and $f: A \to \mathbb{C}$, let us assume $\alpha \in A$.

1. If $\displaystyle \lim_{z \to \alpha } f(z) = f (\alpha)$, then $f$ is said to be continuous at $\alpha$, and if $f$ is continuous at every point in the complex domain $\mathscr{R}$, then $f$ is said to be continuous on $\mathscr{R}$. Especially, if $f$ is continuous throughout its domain, it is called a continuous function¹.

2. The derivative of $f$ at $\alpha$ is defined as following, and if the derivative exists at $\alpha$, $f$ is said to be differentiable at $\alpha$. $$ f ' (\alpha) := \lim_{h \to 0} {{ f ( \alpha + h ) - f ( \alpha ) } \over { h }} $$ Wherein $h \in \mathbb{C}$, it should be independent of the direction in the complex plane.

3. If $f$ is differentiable at every point in the complex domain $\mathscr{R}$, then $f$ is said to be analytic on $\mathscr{R}$. In particular, if $f:\mathbb{C} \to \mathbb{C}$ is analytic at $\mathbb{C}$, it is called an Entire function².

Explanation

2. Unlike functions with the real number set $\mathbb{R}$ as a domain, functions with $\mathbb{C}$ as a domain generally do not carry the same geometric meaning. However, there is no reason why differentiation in complex analysis should not be referred to as differentiation in the formal definition. Of course, if we consider $\mathbb{C} \simeq \mathbb{R}^{2}$ as the complex plane, it can still be seen as having a similar meaning to the slope.
3. Analytic functions are also referred to as Regular Functions or Holomorphic Functions. However, the term “analytic function” is most commonly used, referring to the conditions for analytic continuation. The question “Why are these functions called analytic functions instead of simply differentiable functions?” may include the perspective from the time when complex analysis was developing. As mentioned, the concept of differentiation in the complex plane is just a formal definition and perhaps was meant not to be thought of in the same way as we considered in the real number space $\mathbb{R}$.