# Analyzing isolated singularities using $\lim_{z\to z_0}(z-z_0)f(z)$

If we have a complex function $$f(z)$$ with a simple pole at $$z_0$$ we can use

$$\operatorname{Res}(f;z_0)=\lim_{z\to z_0}(z-z_0)f(z) \tag{1}$$ to find its residue for $$z_0$$. Now, I read that we can also use $$(1)$$ to analyze another isolated singularity $$z_1$$. Basically I read that we can say the following:

• $$(1)=c, c$$ finite: we have a simple pole
• $$(1)=\infty$$: Pole of higher order
• $$(1)=0$$: $$f(z)$$ is analytic at $$z_1$$

Now, all three cases make sense to me, if we assume, that $$z_1$$ is a pole. Sadly, it wasn't clearly stated so I started to think about "What if $$z_1$$ is a removable or essential singularity? Can we still use $$(1)$$ to gather information and if so, what kind of information?"

From how I understand $$(1)$$ and how it is derive and if one thinks about the properties of the Laurent-Series of removable and essential isolated singularities I'd say we can't use $$(1)$$ to really get more information about $$z_1$$ if we assume it to be any kind of isolated singularity.

You should just examine the other two cases. If we have a removable singularity at $$w\in \mathbb{C}$$, then the Laurent Expansion of $$f$$ at $$w$$ is just its Taylor expansion $$f(z)=\sum_{k=0}^\infty c_k(z-w)^k.$$ To this end,
$$\lim_{z\to w} f(z)(z-w)=\lim_{z\to w} \left(\sum_{k=0}^\infty c_k(z-w)^k\right)(z-w)=0.$$ On the other hand, if there is an essential singularity, we Laurent expand as $$f(z)=\sum_{k=-\infty}^\infty c_k(z-w)^k.$$ Then, $$\lim_{z\to w} f(z)(z-w)=\lim_{z\to w} \sum_{k=-\infty}^\infty c_k(z-w)^{k+1}$$ which does not exist by the characterization of isolated essential singularities.
If $$f$$ has an isolated singularity at $$z_0$$, then the limit $$(1)$$ doesn't exist. This follows from the Casorati-Wierstrass theorem. On the other hand, the limit $$(1)$$ doesn't allow you to distinguish the case in which $$f$$ is actually defined at $$z_0$$ (and it's analytic there) from the case in which $$z_0$$ is a removable singularity of $$f$$ (but, of course, you have the definition of $$f$$ in order to distinguish these two cases).