§4.2.1 Isolated Singularities at the $\infty$ Point

Given the one-point compactification of $ℂ$, $\widehat{ℂ}$, we can now define and analyze the behavior of functions near the point at $\infty$. Similar to the classification of isolated singularities in $ℂ$, we can classify $\infty$ as a removable singularity, a pole, or an essential singularity of a holomorphic function.

Let $𝑓:ℂ\setminus \overline{𝐷\left(0,𝑅\right)}\to ℂ$ be holomorphic for some $𝑅>0$. Then $𝑧=\infty$ is an isolated singularity of $𝑓$. To analyze the nature of the singularity, let $𝜁=\frac{1}{𝑧}$. We define a new function $𝑔\left(𝜁\right)=𝑓\left(\frac{1}{𝜁}\right)=𝑓\left(𝑧\right)$, which is holomorphic on ${𝐷}^{\ast }\left(0,\frac{1}{𝑅}\right)$. Then at $𝜁=0$, $𝑔\left(𝜁\right)$ has the Laurent expansion of

where $𝜑$ and $𝜓$ are the holomorphic and principal parts of $𝑔$, respectively. Let $\widetilde{𝜑}\left(𝑧\right)=𝜑\left(\frac{1}{𝑧}\right)$, $\widetilde{𝜓}\left(𝑧\right)=𝜓\left(\frac{1}{𝑧}\right)$. At $𝑧=0$, $𝑓$ then has the Laurent expansion of

The classification of the singularity at $\infty$ is then reduced to the classification of the singularity of $𝑔$ at $0$:

1. If $𝑧=\infty$ is a removable singularity of $𝑓\left(𝑧\right)$, then $𝑓\left(𝑧\right)$ has the form of

   $$𝑓\left(𝑧\right)={𝑐}_0+\frac{{𝑐}_{-1}}{𝑧}+\frac{{𝑐}_{-2}}{{𝑧}^2}+\frac{{𝑐}_{-3}}{{𝑧}^3}+\cdots .$$

2. If $𝑧=\infty$ is a pole of $𝑓\left(𝑧\right)$ with degree $𝑚\in \mathbb{N}$, then $𝑓\left(𝑧\right)$ can be written as

   $$𝑓\left(𝑧\right)={𝑐}_{𝑚}{𝑧}^{𝑚}+{𝑐}_{𝑚-1}{𝑧}^{𝑚-1}+\cdots +{𝑐}_0+\frac{{𝑐}_{-1}}{𝑧}+\cdots ,$$
   where ${𝑐}_{𝑚}\ne 0$.

3. If $𝑧=\infty$ is an essential singularity of $𝑓\left(𝑧\right)$, then $𝑓\left(𝑧\right)$ can be expanded as

   $$𝑓\left(𝑧\right)=\sum _{𝑛=-\infty }^{\infty }{𝑐}_{𝑛}{𝑧}^{𝑛},$$
   where $\forall 𝑁\in \mathbb{N}$, $\exists 𝑛>𝑁$ such that ${𝑐}_{𝑛}\ne 0$ (infinitely many coefficients of $𝜓$ or $\widetilde{𝜓}$ are nonzero).