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Complex analysis question

  1. Feb 18, 2008 #1
    1. The problem statement, all variables and given/known data
    Evaluate [tex]\int_{\gamma} \sqrt {z} dz[/tex] where [tex]\gamma[/tex] 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. jcsd
  3. Feb 18, 2008 #2
    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.
     
  4. Feb 19, 2008 #3
    Anyone know how to prove that the integral is independent of the branch if the branch makes sense?
     
  5. Feb 19, 2008 #4
    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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