9.3 Isolated Singularities

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Taylor's theorem gives us a clear picture of the local structure of a holomorphic function: if $f$ is holomorphic on $\mathsf{B}(b,R)$ then $f$ can be represented by a power series centered at $b$ with radius of convergence at least $R$. Laurent's theorem augments this with a clear picture of structure of a holomorphic function on an annulus: if $f$ is holomorphic on $\mathsf{Ann}(b,r,R)$ then $f$ can be represented by a Laurent series on $\mathsf{Ann}(b,r,R)$. We will now concentrate on the case $r=0$. That is, we will say a bit more about the structure of a holomorphic function on a punctured ball $\mathsf{Ann}(b,0,R)$.

Definition (Isolated Singularity)

An isolated singularity of a function $f$ is a point $b\in \mathbb{C}$ such that $f$ is holomorphic on $\mathsf{Ann}(b,0,R)$ for some $R>0$.

If $b$ is an isolated singularity of $f$ then $f$ is holomorphic on $\mathsf{Ann}(b,0,R)$ so can be represented by a Laurent series $$f(z)=\sum _{n=1}^{\mathrm{\infty }}{a}_n(z-b{)}^{-n}+a_0+\sum _{n=1}^{\mathrm{\infty }}{a}_n(z-b{)}^n$$ on $\mathsf{Ann}(b,0,R)$. We classify the nature of the singularity $b$ according to the principal term. Each of the following are possible.

• No terms.
• Finitely many terms.
• Infinitely many terms.

The point $b$ is then described respectively as follows.

• A removable singularity.
• A pole.
• An essential singularity.

If $b$ is an isolated singularity that is removable then $f$ is equal to the power series $$g(z)=a_0+\sum _{n=1}^{\mathrm{\infty }}{a}_n(z-b{)}^n$$ on $\mathsf{Ann}(b,0,R)$. But, as a power series on $\mathsf{B}(b,R)$ the function $g$ is continuous and differentiable at $b$. This is the sense in which the singularity is removable: if we extend the definition of $f$ to $\mathsf{B}(b,R)$ by declaring that $f(b)=a_0$ we get a function that is holomorphic on $\mathsf{B}(b,R)$.

Example

Define $f:\mathbb{C}\setminus \{0\}\to \mathbb{C}$ by $$f(z)={\displaystyle \frac{\sin (z)}{z}}$$ for all $z\ne 0$. It is not defiend at $0$ and therefore certainly not holomorphic at $0$, but is differentiable at every non-zero complex number, so $0$ is an isolated singularity of $f$. If $z\ne 0$ then $$\begin{array}{rl}{\displaystyle \frac{\sin (z)}{z}}& ={\displaystyle \frac{1}{z}}\sum _{k=0}^{\mathrm{\infty }}{\displaystyle \frac{(-1{)}^k}{(2k+1)!}}{z}^{2k+1}\\ & =\sum _{k=0}^{\mathrm{\infty }}{\displaystyle \frac{(-1{)}^k}{(2k+1)!}}{z}^{2k}\\ & =1-{\displaystyle \frac{z^2}{3!}}+{\displaystyle \frac{z^4}{5!}}-{\displaystyle \frac{z^6}{7!}}+\cdots \end{array}$$ which is a power series with infinite radius of convergence. Thus $0$ is a removable singularity of $f$. If we declare $f(0)=1$ then $f$ is defined on all of $\mathbb{C}$ and is holomorphic there.

When $b$ is an isolated singularity that is a pole we represent $f$ as $$f(z)={\displaystyle \frac{a_{-m}}{(z-b{)}^m}}+\cdots +{\displaystyle \frac{a_{-1}}{(z-b)}}+a_0+\sum _{n=1}^{\mathrm{\infty }}{a}_n(z-b{)}^n$$ on $\mathsf{Ann}(b,0,R)$. We say that $f$ has a pole of order $m$ at $b$ in this case. A pole of order 1 is called a simple pole.

Example

Define $f:\mathbb{C}\setminus \{0\}\to \mathbb{C}$ by $$f(z)={\displaystyle \frac{\sin (z)}{z^4}}$$ for all $z\ne 0$. If $z\ne 0$ we have $$\begin{array}{rl}f(z)& =\frac{1}{z^4}(z-{\displaystyle \frac{z^3}{3!}}+{\displaystyle \frac{z^5}{5!}}-{\displaystyle \frac{z^7}{7!}}+\cdots )\\ & ={\displaystyle \frac{1}{z^3}}-{\displaystyle \frac{1}{3!}}{\displaystyle \frac{1}{z}}+{\displaystyle \frac{z}{5!}}-{\displaystyle \frac{z^3}{7!}}+\cdots \end{array}$$ and read off that $f$ has a pole of order $3$ at $0$.

Often we will be interested in functions $f:D\to \mathbb{C}$ that are holomorphic on $D\setminus \{b_1,\dots ,b_t\}$ and have poles at each of the singularities $b_1,\dots ,b_t$.

Definition (Meromorphic Function)

By a meromorphic function on a domain $D$ we will mean a function $f:D\setminus \{b_1,\dots ,b_t\}$ with poles at each of the points $b_1,\dots ,b_t$.

When $b$ is an isolated singularity that is an essential singularity we represent $f$ as $$f(z)=\sum _{n=1}^{\mathrm{\infty }}{a}_n(z-b{)}^{-n}+a_0+\sum _{n=1}^{\mathrm{\infty }}{a}_n(z-b{)}^n$$ on $\mathsf{Ann}(b,0,R)$ with infinitely many of the coefficients $a_{-n}$ non-zero.

Example

The functions $$\mathsf{exp}(1/z^2)=1+{\displaystyle \frac{1}{z^2}}+{\displaystyle \frac{1}{2!}}{\displaystyle \frac{1}{z^4}}+{\displaystyle \frac{1}{3!}}{\displaystyle \frac{1}{z^6}}+\cdots$$ and $$\sin (1/z)={\displaystyle \frac{1}{z}}-{\displaystyle \frac{1}{3!}}{\displaystyle \frac{1}{z^3}}+{\displaystyle \frac{1}{5!}}{\displaystyle \frac{1}{z^5}}-{\displaystyle \frac{1}{7!}}{\displaystyle \frac{1}{z^7}}+\cdots$$ both have essential singularities at $0$.

One remarkable result about essential singularities is that the image of $\mathsf{Ann}(b,0,R)$ under $f$ is the punctured plane for each $R>0$ whenever $f$ has an essential singularity at $b$. (This is called the great Picard theorem.) Essential singularities are difficult to handle and we will say no more about them in this course.