Maybe the question is trivial, but I cannot find an answer according to the standard books of Complex Analysis. We have three kinds of singularities: isolated, pole, and essential.
What kind of singularities are the following ones?
1) $\frac{1}{\sqrt z} \text{ in } z=0$
2) $\frac{1}{\sqrt z-1} \text{ in } z=1$
One could make up a "fractionary order" pole, but they are not provided in usual terminology, since the Laurent series is on integer numbers... The problem is that they actually do not admit any Laurent expansion in the foregoing points, and thus one cannot say that they are essential singularities and apply the Picard theorem....
So what's the truth?