How can we find the pole of square root function? What is the pole of the following function
$$f(z)=\frac{\pi}{1+\sqrt{z}}, \quad z\in C$$

There is no "pole", but there is a "branch point" at $z=0$. Looking at the monodromy of the function around $z=0$, we see that the values change as $\sqrt{z}$ changes by $-1$. It doesn't "blow up" at $z=0$...
