I'm asked to find residue of the form $\text{exp}(\frac{1}{1+\sqrt{z}})\text{d}z$ at the point 1. It's clear that this function is multivalued and we have to make branch cut. Let us cut along $(-\infty, 0)$. First branch is determined by choosing $\sqrt 1 = 1$ and second branch by choosing $\sqrt 1 = -1$.
In the first case everything is clear: function $\text{exp}(\frac{1}{1+\sqrt{z}})$ is regular at 1, so residue is equal to 0.
The second case is a problem. I can show that $\text{exp}(\frac{1}{1+\sqrt{z}})$ has essential singularity at point 1. I tried to use series expansion of exp to find residue from Laurent series, but it led me nowhere.
Any hints? Thanks!