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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?

Arctic Char
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TOTAN GHOSH
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1 Answers1

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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 $\{z\in \mathbb{C}:g(z)\in D\}=g^{-1}(D)$.

Masacroso
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  • 1st of all thanks for looked onto it. – TOTAN GHOSH Sep 25 '21 at 14:13
  • But my question is what is the branch cut? I mean here the algebraic branch points are $a,b,c$ but is the branch cut? Is the function is analytic on $\mathbb{C}-$the triangle formed by the points $a, b, c$ – TOTAN GHOSH Sep 25 '21 at 14:17
  • @TOTAN take a look here to understand the concept of branch cut. For your second comment, please read again my answer. – Masacroso Sep 25 '21 at 14:34
  • Yess, I got it, ... Thanks again – TOTAN GHOSH Sep 25 '21 at 14:45
  • Note that not all branches can be constructed in that way. As a simpler example, an analytic branch of $\sqrt {z^2 - 1}$ on $|z| > 1$ is an odd function, so it cannot be written as $(z \mapsto \sqrt z) \circ (z \mapsto z^2 - 1)$ for any choice of $\sqrt z$. – Maxim Sep 25 '21 at 19:26
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    @TOTANGHOSH There isn't an analytic branch in the exterior of $\triangle abc$, because $\infty$ is also a branch point of $\sqrt {(z - a) (z - b) (z - c)}$. – Maxim Sep 25 '21 at 19:32