Infinite Limits of Complex Functions

Infinite Limits

Notation: Infinite Limits

Let $f:\mathcal{N}\subseteq \mathbb{C}\to \mathbb{C}$ be a complex function defined on some deleted neighborhood of $c\in \mathbb{C}$.

We write

$$\underset{z\to c}{\lim }f(z)=\infty$$

if, for each $M>0$, there exists some $\delta >0$ such that

$$0<|z-c|<\delta ⟹|f(z)|>M\forall z\in \mathcal{N}.$$

Characterizations

Theorem: Infinite Limit via Modulus

Let $f:\mathcal{N}\subseteq \mathbb{C}\to \mathbb{C}$ be a complex function defined on some deleted neighborhood of $c\in \mathbb{C}$.

The limit of $f$ for $z\to c$ is $\infty$ if and only if the limit of $|f|$ for $z\to c$ is $\infty$.

$$\underset{z\to c}{\lim }f(z)=\infty ⟺\underset{z\to c}{\lim }|f(z)|=\infty$$

PROOF

TODO

Infinite Limits at Infinity

Notation: Limits at Infinity

Let $f:\mathcal{D}\subseteq \mathbb{C}\to \mathbb{C}$ be a complex function such that $\mathcal{D}$ is the complement of some open ball around $0$.

We write

$$\underset{z\to \infty }{\lim }f(z)=\infty$$

if for each $M>0$ there exists some $R>0$ such that

$$|z|>R⟹|f(z)|>M\forall z\in \mathcal{D}.$$

WARNING

Even though we use limit notation for infinite limits and infinite limits at infinity, we never say that these limits exist, since they are not complex numbers.

Characterizations

Theorem: Infinite Limit via Absolute Value

Let $f:\mathcal{D}\subseteq \mathbb{C}\to \mathbb{C}$ be a complex function such that $\mathcal{D}$ is the complement of some open ball around $0$.

The limit of $f$ for $z\to \infty$ is $\infty$ if and only if the limit of $|f|$ for $|z|\to \infty$ is $\infty$.

$$\underset{z\to \infty }{\lim }f(z)=\infty ⟺\underset{|z|\to \infty }{\lim }|f(z)|=\infty$$

PROOF

TODO

Converting Limits

Theorem: Infinite Limit $\leftrightarrow$ Infinite Limit at Infinity

Let $f:\mathcal{D}\subseteq \mathbb{C}\to \mathbb{C}$ be a complex function such that $\mathcal{D}$ is the complement of some open ball around $0$.

The limit of $f$ for $z\to \infty$ is $\infty$ if and only if the limit of $f\left(\frac{1}{z}\right)$ for $z\to 0$ is $\infty$.

$$\underset{z\to \infty }{\lim }f(z)=\infty ⟺\underset{z\to 0}{\lim }f\left(\frac{1}{z}\right)=\infty$$

PROOF

TODO

Bibliography

1. N. H. Asmar, L. Grafakos, “Analytic Functions,” in Complex Analysis with Applications, Columbia, USA: Springer, 2018