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

[ehrenfest]

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

 

[Mystic998]

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.

 

[ehrenfest]

Anyone know how to prove that the integral is independent of the branch if the branch makes sense?

 

[Mystic998]

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.