- #1

mahler1

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**Homework Statement .**

Let ##n \in \mathbb N##. If ##\Omega \subset \mathbb C^*## is open, we define a branch of the nth-root of ##z## on ##\Omega## to be any continuous function ##f:\Omega \to \mathbb C## such that ##{f(z)}^n=z## for all ##z \in \Omega##. We will denote ##\sqrt[n]{z}## to ##f(z)##.

(i) Prove that if ##\Omega=\mathbb C \setminus \mathbb R_{\leq 0}##, there are exactly two branches of ##\sqrt{z}## on ##\Omega##. Define them. Show that every branch of ##\sqrt{z}##

is holomorphic.

(ii) If ##\Omega## is connected and ##f## is a branch of ##\sqrt{z}## on ##\Omega##, then ##f## and ##-f## are all the branches.

**The attempt at a solution**

For ##(i)##

By definition, ##f(z)^2=e^{2\log(f(z))}##. This means ##e^{2log(f(z))}=z##, So ##2log(f(z))## is a branch of the logarithm on ##\Omega##. I am stuck at that point.

And I also don't know how to deduce that if ##\Omega=\mathbb C \setminus \mathbb R_{\leq 0}##, then there are two functions ##f_1## and ##f_2## that satisfy the conditions required. For ##(ii)## I have no idea where to start the problem, I would appreciate help and suggestions.

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