Proving Nth-Root Branches on Open Sets

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In summary: How does that lead to the conclusion that there are exactly two branches?In summary, 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##. For part (i), we can use the definition of the branch of logarithm to show that there are exactly two branches of ##\sqrt{z}## on ##\Omega=\mathbb C \setminus \mathbb R_{\leq 0}##. For part (ii), we can show that if there are two different values of ##f(z)##
  • #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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  • #2
mahler1 said:
we call a branch of the nth-root of ##z## on ##\Omega## to every continuous function ##f:\Omega \to \mathbb C## such that ##{f(z)}^n=z## for all ##z \in \Omega##.
I guess you mean
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##.
By definition, ##f(z)^2=e^{2f(z)}##.
Do you mean ##f(z)^2=e^{2\ln(f(z))}##?
 
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  • #3
haruspex said:
I guess you mean
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##.

Do you mean ##f(z)^2=e^{2\ln(f(z))}##?

Thanks for the corrections and sorry for my english. I've edited my original post.
 
  • #4
Sorry for the delay...
Suppose there are two different values w, w' of f(z). Both satisfy w2=z. What can you say about the relationship between them?
 

FAQ: Proving Nth-Root Branches on Open Sets

What is the purpose of proving nth-root branches on open sets?

The purpose of proving nth-root branches on open sets is to establish the existence and uniqueness of nth-roots on open sets in mathematical analysis. This is important in understanding and solving equations involving nth-roots, as well as in further applications of these concepts.

What are nth-roots?

Nth-roots are mathematical operations that find the number which, when multiplied by itself n times, gives a given number. For example, the 3rd root of 27 is 3, as 3 x 3 x 3 = 27. Nth-roots are commonly denoted as √n or n√.

What are open sets?

Open sets are subsets of a metric space (a set of points with a defined distance function) that do not contain their boundary points. In other words, for any point in an open set, there exists a small enough neighborhood around that point that is entirely contained within the set. Open sets are commonly used in topology, a branch of mathematics that deals with the properties of spaces.

How are nth-root branches proven on open sets?

Nth-root branches can be proven on open sets using various mathematical techniques, such as the intermediate value theorem, the mean value theorem, and the inverse function theorem. These techniques involve analyzing the behavior of the nth-root function on the open set and proving its continuity, differentiability, and invertibility.

What are the implications of proving nth-root branches on open sets?

Proving nth-root branches on open sets has important implications in various fields of mathematics, including calculus, analysis, and topology. It allows for the accurate calculation and manipulation of nth-roots on open sets, and also provides a foundation for more complex mathematical concepts and applications.

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