I need to find the singularities of
$$f(z) = \frac{1-e^z}{2+e^z}$$
My effort: Poles of function are given by
$$2+e^z=0\implies e^z = -2 \implies z = \log 2+i(2k+1)\pi$$ for k integer.
All these are singularities termed as simple poles. By definition, limit point of these which is $\infty$ is a non-isolated singularity.
Further, limit point of zeros is again infinity which is a isolated-essential singularity.
But if both of isolated and non isolated coincides we take it as a non-isolated singularity. Am i correct? These are the only singularities?
Best Answer
Every neighborhood of $\infty$ contains a pole, implying, as you correctly state, that $\infty$ is not an isolated singularity. What makes you believe that it is also an isolated singularity?