Branch cut and branch points Recently I am studying branch points and branch cuts in complex analysis.
If there is a function like $f(z)=\sqrt{(z-1)(z-2)}$, then the algebraic branch points are $1$ and $2$ and the branch cut is the interval $[1 , 2]$.
One can ask a natural question: Let $f(z)=\sqrt{(z-a) (z-b) (z-c)}$ where $a, b, c$ are three different non zero complex numbers. How to find the branch points and branch cut?

Let $s(z):=\sqrt{z}$. Then if $g(z):=(z-a)(z-b)(z-c)$ you have that $g$ is a polynomial, so it is an entire function, however you must choose a branch of the function $s$, as it is not continuous in the whole $\mathbb{C}$.
Choosing a branch of $s$ is the same thing than choosing a branch of the complex logarithm, as we generally defines $\sqrt{z}:=\exp\left(\frac1{2}\log z\right)$ for some branch of the complex logarithm. Setting some branch of the logarithm we will have a domain $D$ where our logarithm is analytic, therefore the domain of $s\circ g$ will be $\{z\in \mathbb{C}:g(z)\in D\}=g^{-1}(D)$.
