Complex Analysis/Isolated singularity

Definition

Let $G\subseteq ℂ$ be a domain and $z_0\in G$. If $f:G\setminus z_0\to ℂ$ is a holomorphic function, then $z_0$ is called an isolated singularity of $f$.

Classification

Depending on the behavior of $f$ in the neighborhood of $z_0$, one distinguishes three different types of isolated singularities of $f$.

Removable Singularities

If $f$ can be holomorphically extended to the entire domain $G$, then we say that $z_0$ is a removable singularity. According to the Riemann Removability Theorem, this is the case if $f$ is bounded in a neighborhood of $z_0$.

Poles

If $z_0$ is not a removable singularity, but there exists an $n\ge 1$ such that $(\cdot -z_0{)}^n\cdot f$ has a removable singularity at $z_0$, then we say that $f$ has a pole at $z_0$. The smallest such $n$ is called the order of the pole.

Essential Singularities

If $z_0$ is neither removable nor a pole, then $z_0$ is called an essential singularity of $f$.

Examples

• Since $\underset{z\to 0}{\lim }\frac{\sin z}{z}=1$, the function $f_1(z)=\frac{\sin z}{z}$ has a removable singularity at $z_0=0$.
• The function $f_2(z)=\frac{1}{\sin z}$ does not have $z_0=0$ a removable singularity at, since$f_2$ is unbounded at $0$, but $f_2$ has a first-order pole at $0$, because $f_2(z)\cdot (z-0{)}^1=f_2(z)z=\frac{z}{\sin z}$ and $\underset{z\to 0}{\lim }\frac{z}{\sin z}=1$, which has a removable singularity at 0 .

• The function $f_3(z)=\sin \frac{1}{z}$ has an essential singularity at $z_0=0$, since for every $n\ge 1$, the function $f_3(z)z^n=z^n\sin \frac{1}{z}$ is unbounded in any neighborhood of $0$. To see this, consider$\sin z^{-1}=\frac{e^{i{z}^{-1}}-e^{-i{z}^{-1}}}{2i}$.For $z=it$ with $t\in ℝ$ is also $f_3(it)(it{)}^n=(it{)}^n\frac{e^{t^{-1}}-e^{-t^{-1}}}{2i}$,which diverges as $t\to 0^{+}$ .

Laurent Expansions

The type of isolated singularity can also be inferred from the Laurent Expansion of

$f$

around

$z_0$

. Let

$f(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_n(z-z_0{)}^n$

be the Laurent Series of $f$ around $z_0$. We define

$o_z(f)=\sup \{n\in \mathbb{Z}|\mathrm{\forall }k<n:a_k=0\}$.

Then, $f$ has the following singularities:

• $o_z(f)\ge 0$, i.e., all negative coefficients vanish, the main part of the series is zero, and the singularity is removable.

• $-\mathrm{\infty }<o_z(f)<0$, i.e., only finitely many negative coefficients are nonzero, there is a pole of order $-o_z(f)$.
• $o_z(f)=-\mathrm{\infty }$, i.e., infinitely many negative coefficients are nonzero, the singularity is essential.

Examples

Let us consider our three examples again:

• It is $f_1(z)=\frac{\sin z}{z}=\sum _{k=0}^{\mathrm{\infty }}(-1{)}^n\frac{z^{2n}}{(2n+1)!}$, so $o_0(f_1)=0$, a removable singularity.

• It is $f_2(z)=\frac{1}{\sin z}=\frac{1}{z}+\frac{z}{6}+\frac{7}{360}{z}^3+\dots$ so $o_0(f_2)=-1$, a pole of first order.

• It is $f_3(z)=\sin z^{-1}=\sum _{n=-\mathrm{\infty }}^0\frac{(-1{)}^n}{(-2n+1)!}{z}^{2n-1}$, so $o_0(f_3)=-\mathrm{\infty }$, an essential singularity.

Page information

Translation and Version Control

This page was translated based on the following Wikiversity source page and uses the concept of Translation and Version Control for a transparent language fork in a Wikiversity:

• Source: Kurs:Funktionentheorie/isolierte Singularität -URL:

https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/isolierte_Singularität

• Date: 11/20/2024