What I know:
I understand that for a complex function to be a multi-valued funciton they must have some branch points. I understand branch points as this definition (translated into English from my textbook):
Suppose that $w = f(z)$. $z_0$ is a branch point if and only if $\exists r \gt 0$, such that when $z$ rotates around the circle $|z_0-z|\lt r$, $w$ does not return to its original value, and that when $\lim_{r\to 0}$, $w$ still does not return to its original value.
Methods of Mathematical Physics, 3rd edition, Peking University Press, p.22
In simple compelx functions like $f(z) = \sqrt{z-a}$, it is straightforward to apply the above definition in the complex plane and identify that one of $z_0$ is $z_0 = a$ and thus deduce $$ |w|=\sqrt{|z-a|}, \arg w = \frac{1}{2}\arg(z-a) $$ such that when $z$ rotates around $a$ for $2\pi$, $w$ only rotates by $\pi$, thus the multi-value property.
What I don't know:
- How to determine if a more complicated complex function is a multi-valued function?
- How to find branch points in general
Some concrete examples to what I don't know:
- How to determine if $z+\sqrt{z-1}$ and $\frac{\sin{\sqrt{z}}}{\sqrt{z}}$ are multi-valued functions?