📂Complex Anaylsis

Limits of Complex Functions

Definition ¹

Let $f : \mathbb{C} \to \mathbb{C}$ be a complex function $f : A \to \mathbb{C}$ defined on an open set $A \subset \mathbb{C}$ where $\alpha \in \overline{A}$. To say that $f(z)$ converges to the limit $l$ when $z \to \alpha$ means for all $\varepsilon > 0$, there exists $\delta > 0$ such that $$ 0 < \left| z - \alpha \right| < \delta \implies \left| f(z) - l \right| < \varepsilon $$ is satisfied, and is denoted as follows. $$ \lim_{z \to \alpha} f(z) = l $$

Properties

Assume both $\lim_{z \to \alpha} f(z)$ and $\lim_{z \to \alpha} g(z)$ exist.

• Uniqueness: If $\displaystyle \lim_{z \to \alpha} f(z)$ exists, then it is unique.
• Conjugate: $$\lim_{z \to \alpha} \overline{z} = \overline{\alpha}$$
• Multiplication by a constant: For all $k \in \mathbb{C}$, $$ \lim_{z \to \alpha} k f(z) = k \lim_{z \to \alpha} f(z) $$
• Addition: $$\lim_{z \to \alpha} \left[ f(z) + g(z) \right] = \lim_{z \to \alpha} f(z) + \lim_{z \to \alpha} g(z)$$
• Multiplication: $$\lim_{z \to \alpha} f(z) g(z) = \lim_{z \to \alpha} f(z) \lim_{z \to \alpha} g(z)$$
• Division: Only when $\displaystyle \lim_{z \to \alpha} g(z) \ne 0$, $$\lim_{z \to \alpha} {{ f(x) } \over { g(x) }} = {{ \lim_{z \to \alpha} f(z) } \over { \lim_{z \to \alpha} g(z) }}$$

Explanation

Note that it was not necessary for $\alpha$ to specifically belong to the domain of $f$ in the definition, and it was never stated that $l = f(\alpha)$.

The difference between a Real Valued Function and a Complex Valued Function is in their direction. While $\mathbb{R}$ to $x \to a$ feels like approaching from either side in $a$, approaching $z \to \alpha$ from $\mathbb{C}$ feels like coming from all directions on the complex plane.