Singularities of $\frac{1}{e^{\frac{1}{z}}+2}$ and classification?

Question

I'm considering the function

$$\frac{1}{e^{\frac{1}{z}}+2}$$

Clearly, it has a unique singularity at $z=0$. I feel like its an essential but I can't find the Laurent expansion or any other way of proving it... Can you help me?

Thank you very much, I usually see these, but this time I'm stuck!

Answer 1

Clearly, it has a unique singularity at $z=0$.

No, that is not the unique singularity. The function has poles at the zeros of

$$e^{\frac{1}{z}} + 2.$$

These poles accumulate at $0$, hence $0$ is not an isolated singularity of the function, and there is no Laurent expansion in any punctured disk $\{ z : 0 < \lvert z\rvert < \rho\}$.

Whether $0$ is classified as an essential singularity depends on whether only isolated singularities are classified such, or all singularities that are neither removable nor poles.

Answer 2

With some imagination, one can see the poles collecting near the origin.

Conformal map of $$ f(z) = \frac{1}{\exp \left( 1 / z \right) + 2}. $$

Blue: $\Re (f)$, Red: $\Im (f)$