Is a pole of fractional order just another name for a certain type of branch point?
The function $\frac{1}{\sqrt{z}}$ has a logarithmic algebraic branch point at $z=0$.
But according to Wolfram Alpha, $ \frac{1}{\sqrt{z-1}}$ has a pole of order $\frac{1}{2}$ at $z=1$.
EDIT:
When asked, Wolfram Alpha still says that $\frac{1}{\sqrt{z-1}}$ has a pole of order $\frac{1}{2}$ at $z=1$.
But now at least it also says that $\frac{1}{\sqrt{z-1}}$ has a branch point there.