📂Complex Anaylsis

Types of Singularities in Complex Analysis

Definition

Singular Point ¹

1. If a function $f$ is differentiable at every point of some $\mathcal{N}(\alpha)$ at $\alpha$, then $f$ is said to be analytic at $\alpha$.
2. If a function $f$ is not analytic at $\alpha \in \mathbb{C}$ but is analytic at some point of every $\mathcal{N}(\alpha)$, then $\alpha$ is called a singular point of $f$.
3. If there exists a $\mathcal{N}(\alpha)$ in which a singular point $\alpha$ is analytic at every point except $\alpha$, then $\alpha$ is said to be isolated.

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• $\mathcal{N}$ denotes a neighborhood, meaning an open set containing $\alpha$.

Types

Let $\alpha \in \mathbb{C}$ be a singular point of $f$.

4. $\displaystyle \exists \lim_{z \to \alpha} f(z) \iff$ $\alpha$ is a removable singular point.
5. $\displaystyle \lim_{z \to \alpha} (z - \alpha)^n f(z) = k \ne 0 \iff$ $\alpha$ is a pole of Order $n$.
6. $\alpha$ is not a pole or is associated with a branch. $\iff$ $\alpha$ is an essential singular point.

Explanation

A pole, in particular when $n=1$, is called a simple Pole.

In fact, unless it is a very perverse case, the point where $f$ is undefined usually becomes the singular point.

For example, if $\displaystyle f(z) = {{z - i} \over {(z^2+1)(z+i)}}$, then the singular points would be $z= \pm i$. They need not be finite either; in the case of $\csc z$, all of $z = n \pi ( n \in \mathbb{Z} )$ are singular points. Meanwhile $\text{Log} z$ has a singular point at $z= 0$, but you may feel it is a little different from the examples given above.

• In $\displaystyle f(z) = {{z - i} \over {(z^2+1)(z+i)}}$, $z = i$ is removable, and $z = -i$ is a pole of order $2$.

• Since $\displaystyle \lim_{z \to n \pi} {{ z - n \pi } \over {\sin z }} = 1$, the singular points of $\csc z$ are all poles of order $1$, that is, simple poles.

• Lastly, in $\text{Log} z$, $z = 0$ is a branch point, so it is an essential singular point.

Such a classification of singular points may at first glance seem like a meaningless game of definitions, but it becomes a very important concept in the later discussion of integration.