Is the Complex Analysis Problem with \(\sqrt{z}\) on the Unit Circle Ambiguous?

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Homework Help Overview

The discussion revolves around evaluating the integral \(\int_{\gamma} \sqrt{z} dz\) where \(\gamma\) is defined as the upper half of the unit circle. Participants are questioning the ambiguity of the problem due to the lack of specification regarding the branch of the complex square root function to use.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • One participant contends that the problem is ambiguous due to the unspecified branch of the square root function. Others suggest that the integral's definition relies on the continuity of the function along the chosen path and question how to prove the independence of the integral from the branch selected.

Discussion Status

The discussion is ongoing, with participants exploring the implications of branch choice on the integral's evaluation. Some guidance has been offered regarding the relationship between the existence of an analytic antiderivative and the integral's independence from branch selection.

Contextual Notes

There is a noted concern about the ambiguity stemming from the problem's lack of clarity on the branch of the square root function, which may affect the integral's evaluation.

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Homework Statement


Evaluate \int_{\gamma} \sqrt {z} dz where \gamma is the upper half of the unit circle.

I contend that this problem does not make sense i.e it is ambiguous because they did not tell us specifically what branch of the complex square root function to use. Am I right?

Homework Equations


The Attempt at a Solution

 
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Well, the integral isn't defined if the function fails to be continuous on the path. So, while you could say, "It's too ambiguous," it's pretty clear that they mean, "Pick a branch for which this makes sense." Now, if the integral winds up depending on the branch you choose for which it makes sense, maybe you've got a problem. But I'm pretty sure it wouldn't.

Though don't quote me on that. I suck at this stuff for some reason.
 
Anyone know how to prove that the integral is independent of the branch if the branch makes sense?
 
Well, that has to do with the function having analytic antiderivative in a region containing the path. Can't remember the proof off the top of my head though.
 

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