Can I Choose Any Branch for the Derivative of sqrt(z) in the Principal Branch?

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In summary, the conversation discusses the uncertainty surrounding the use of multivalued functions and choosing branches. The main question is whether the use of different branches can affect the outcome of calculations, particularly in the case of differentiating $\sqrt{z}$ in the principal branch. The motivation for this question is related to the use of the Christoffel-Schwarz transformation, where the computation is typically done with respect to the principal branch. However, some online examples seem to combine square roots without considering branches, leading to a discrepancy in the results. The issue of analyticity is also raised, as using the principal branch may not always be appropriate in certain regions.
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kcuf
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I understand the basic definitions and ideas behind choosing branches of multivalued functions, but I still have some uncertainty when dealing with multivalued functions. As an example, if I'm in the the principal branch, when can I say that $(z-1)^\frac 1 2(z+1)^\frac 1 2= (z^2-1)^\frac 1 2$; or, what happens when I differentiate $\sqrt z$ in the principal branch, the resulting derivative is $z^{-\frac 1 2}$, am I free to choose whatever branch I want for this derivative, or does it have to correspond to that of the original function.

If anyone could help me clarify this, I would greatly appreciate it.

Thanks
 
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also, maybe I'm just an idiot, but I how do I get this forum to display tex?
 
  • #3
Maybe I should clarify what's motivating this question:
I'm trying to use the Christoffel-Schwarz, and the text I have been using says that the powers are computed with respect to the principal branch, however examples that I have seen online just seem to naively combine squareroots together in order to find a potential antiderivative, for example http://planetmath.org/encyclopedia/ExampleOfSchwarzChristoffelTransformation.html [Broken]. What also troubles me with this example, is if we were restricted to the principal branch, then $\sqrt{z^2-1}$ is not analytic on the upper half plane (because it sends most of the imaginary axis to the negative real axis), so it shouldn't be an antiderivative unless we can choose a different branch for the derivative (which seems wierd), right?
 
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1. What are the different branches of squareroot?

There are two branches of squareroot, the principal square root (also known as the positive square root) and the negative square root.

2. How are the branches of squareroot represented?

The principal square root is represented by the symbol √ and the negative square root is represented by -√.

3. What is the difference between the principal square root and the negative square root?

The principal square root gives the positive solution to a square root equation, while the negative square root gives the negative solution.

4. When do we use the principal square root and when do we use the negative square root?

The principal square root is used when we are looking for the positive solution to an equation, while the negative square root is used when we are looking for the negative solution.

5. Can the branches of squareroot be simplified?

Yes, the branches of squareroot can be simplified by finding the perfect square factors within the radical and taking them out as a single number. However, this can only be done for the principal square root, as the negative square root cannot be simplified.

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