The title pretty much explains it. I'm trying to answer a question where I'm given a few complex functions and it asks me to identify their singularities, and then to classify any that are isolated.
The one I'm having trouble with is
$$\frac{1}{\exp\left(\frac{1}{z}\right) + 2}$$
I'm pretty sure $z=0$ is a singularity (since the function is undefined here), but I have no idea how to see if there are any others. I know the complex exponential can take any non-zero value – but I have no idea how to solve for $\exp(\frac{1}{z}) = -2$, much less how to classify them.
Your help would be really appreciated!
Best Answer
Well, $0$ is a singularity, yes. A non-isolated one.
Besides consider the equation $e^w+2=0$. Its solutions are the numbers of the form $\log2+k\pi i$, where $k$ is an odd integer. So, $\exp\left(\frac1z\right)+2=0$ if and only if $z=\frac1{\log2+k\pi i}$, for some odd integer. These are (together with $0$) the singularities of your function. And suppose that it is clear now why $0$ is a non-isolated singularity. All other singularities are isolated ones. And all of them are simple poles (since they are all simple zeros of $\exp\left(\frac1z\right)+2$).