Limits of Complex Functions

Limits of Complex Functions

The concept of a limit of a complex function is analogous to that of a limit of a real function. We define this concept below.

Definition: Let $A \subseteq \mathbb{C}$ and let $z_0 \in \mathbb{C}$ be an accumulation point of $A$. The Limit of $f$ as $z$ Approaches $z_0$ is $A$ denoted $\displaystyle{\lim_{z \to z_0} f(z) = A}$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $z \in A$ and $\mid z - z_0 \mid < \delta$ then $\mid f(z) - A \mid < \epsilon$.

A point $z_0 \in \mathbb{C}$ is said to be an accumulation point of $A \subseteq \mathbb{C}$ if for all $r > 0$ we have that $B(z_0, r) \cap A \setminus \{ z_0 \} \neq \emptyset$. The requirement that $z_0$ is an accumulation point of $A$ in the definition above ensures us that we can actually approach $z_0$ in the domain.

For example, consider the function $f(z) = z$ which is the identity function. We claim that for all $z_0 \in \mathbb{C}$ that:

(1)
\begin{align} \quad \lim_{z \to z_0} f(z) = z_0 \end{align}

To prove this, let $\epsilon > 0$ be given. Then notice that:

(2)
\begin{align} \quad \mid f(z) - z_0 \mid = \mid z - z_0 \mid \quad (*) \end{align}

So choose $\delta = \epsilon > 0$. Then if $\mid z - z_0 \mid < \delta$ we have by $(*)$ that:

(3)
\begin{align} \quad \mid f(z) - z_0 \mid < \delta = \epsilon \end{align}

Therefore $\displaystyle{\lim_{z \to z_0} f(z) = z_0}$.

We will now state some basic properties of limits of complex functions that the reader should be familiar with for real functions. The proofs of these theorems are pretty much identical to that for real functions, so we will omit their proofs for now.

Theorem 1 (Uniqueness of Limits): If $\displaystyle{\lim_{z \to z_0} f(z) = A}$ and $\displaystyle{\lim_{z \to z_0} f(z) = B}$ then $A = B$.
Thereom 2: If $\displaystyle{\lim_{z \to z_0} f(z) = A}$ and $\displaystyle{\lim_{z \to z_0} g(z) = B}$ then:
a) $\displaystyle{\lim_{z \to z_0} [f(z) + g(z)] = A + B}$.
b) $\displaystyle{\lim_{z \to z_0} [f(z) - g(z)] = A - B}$.
c) $\displaystyle{\lim_{z \to z_0} f(z)g(z) = A \cdot B}$.
d) $\displaystyle{\lim_{z \to z_0} \frac{f(z)}{g(z)} = \frac{A}{B}}$ provided that $B \neq 0$.
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