I seem to have a mental block regarding branch points...I thought that the singularities of a function determined its branch points but then I read that they are irrelevant when deciding if a point is a branch point of some function? I.e. $\frac{1}{z^2}$ has a singularity at $0$ but it has no branch points.
Suppose $f(z) = z^\frac{1}{2}$. Then $z_0 = 0$ is a branch point for $f(z)$.
But suppose we had $f(z) = (z + i)^{\frac{1}{2}}$. Why is $z_0 = 0$ not a branch point for $f(z)$ in this case? What is it about $f(z)$ that determines it's branch point(s)? How can I determine the branch points for $f(z) = (z + i)^{\frac{1}{2}}$