Complex analysis question

1. Feb 18, 2008

ehrenfest

1. The problem statement, all variables and given/known data
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?

2. Relevant equations

3. The attempt at a solution

2. Feb 18, 2008

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.

3. Feb 19, 2008

ehrenfest

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

4. Feb 19, 2008

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.