Complex Analysis/Isolated singularity

Let ${\displaystyle G\subseteq \mathbb {C} }$ be a domain and ${\displaystyle z_{0}\in G}$. If ${\displaystyle f\colon G\setminus {z_{0}}\to \mathbb {C} }$ is a holomorphic function, then ${\displaystyle z_{0}}$ is called an isolated singularity of ${\displaystyle f}$.

Depending on the behavior of ${\displaystyle f}$ in the neighborhood of ${\displaystyle z_{0}}$, one distinguishes three different types of isolated singularities of ${\displaystyle f}$.

If ${\displaystyle f}$ can be holomorphically extended to the entire domain ${\displaystyle G}$, then we say that ${\displaystyle z_{0}}$ is a removable singularity. According to the Riemann Removability Theorem, this is the case if ${\displaystyle f}$ is bounded in a neighborhood of ${\displaystyle z_{0}}$.

If ${\displaystyle z_{0}}$ is not a removable singularity, but there exists an ${\displaystyle n\geq 1}$ such that ${\displaystyle (\cdot -z_{0})^{n}\cdot f}$ has a removable singularity at ${\displaystyle z_{0}}$, then we say that ${\displaystyle f}$ has a pole at ${\displaystyle z_{0}}$. The smallest such ${\displaystyle n}$ is called the order of the pole.

If ${\displaystyle z_{0}}$ is neither removable nor a pole, then ${\displaystyle z_{0}}$ is called an essential singularity of ${\displaystyle f}$.

• Since ${\displaystyle \lim _{z\to 0}{\frac {\sin z}{z}}=1}$, the function ${\displaystyle f_{1}(z)={\frac {\sin z}{z}}}$ has a removable singularity at ${\displaystyle z_{0}=0}$.
• The function ${\displaystyle f_{2}(z)={\frac {1}{\sin z}}}$ does not have ${\displaystyle z_{0}=0}$ a removable singularity at, since${\displaystyle f_{2}}$ is unbounded at ${\displaystyle 0}$, but ${\displaystyle f_{2}}$ has a first-order pole at ${\displaystyle 0}$, because ${\displaystyle f_{2}(z)\cdot (z-0)^{1}=f_{2}(z)z={\frac {z}{\sin z}}}$ and ${\displaystyle \lim _{z\to 0}{\frac {z}{\sin z}}=1}$, which has a removable singularity at 0 .

• The function ${\displaystyle f_{3}(z)=\sin {\frac {1}{z}}}$ has an essential singularity at ${\displaystyle z_{0}=0}$, since for every ${\displaystyle n\geq 1}$, the function ${\displaystyle f_{3}(z)z^{n}=z^{n}\sin {\frac {1}{z}}}$ is unbounded in any neighborhood of ${\displaystyle 0}$. To see this, consider${\displaystyle \sin z^{-1}={\frac {e^{iz^{-1}}-e^{-iz^{-1}}}{2i}}}$.For ${\displaystyle z=it}$ with ${\displaystyle t\in \mathbb {R} }$ is also ${\displaystyle f_{3}(it)(it)^{n}=(it)^{n}{\frac {e^{t^{-1}}-e^{-t^{-1}}}{2i}}}$,which diverges as ${\displaystyle t\to 0^{+}}$ .

The type of isolated singularity can also be inferred from the Laurent Expansion of ${\displaystyle f}$ around ${\displaystyle z_{0}}$. Let

${\displaystyle f(z)=\sum _{n=-\infty }^{\infty }a_{n}(z-z_{0})^{n}}$

be the Laurent Series of ${\displaystyle f}$ around ${\displaystyle z_{0}}$. We define

${\displaystyle o_{z}(f)=\sup\{n\in \mathbb {Z} |\forall k<n:a_{k}=0\}}$.

Then, ${\displaystyle f}$ has the following singularities:

• ${\displaystyle o_{z}(f)\geq 0}$, i.e., all negative coefficients vanish, the main part of the series is zero, and the singularity is removable.

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

Let us consider our three examples again:

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

• It is ${\displaystyle f_{2}(z)={\frac {1}{\sin z}}={\frac {1}{z}}+{\frac {z}{6}}+{\frac {7}{360}}z^{3}+\ldots }$ so ${\displaystyle o_{0}(f_{2})=-1}$, a pole of first order.

• It is ${\displaystyle f_{3}(z)=\sin z^{-1}=\sum _{n=-\infty }^{0}{\frac {(-1)^{n}}{(-2n+1)!}}z^{2n-1}}$, so ${\displaystyle o_{0}(f_{3})=-\infty }$, an essential singularity.

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