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Math Help - branch cut problem

  1. #1
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    branch cut problem

    Attached Thumbnails Attached Thumbnails branch cut problem-1234.png  
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  2. #2
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    What do you have so far?
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  3. #3
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    re

     \Delta_{C}{arg f}=1,-1,0,0 for (i)(ii)(iii)(iv) respectively.

    how about the behaviour
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  4. #4
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    I'll try (i) for \alpha=1/2:

    You can let z=1+se^{i\phi} and z=re^{i\theta} then \sqrt{\frac{re^{i\theta}}{se^{i\phi}}}=\left(\frac  {r}{s}\right)^{1/2}e^{i/2(\theta-\phi+2k\pi)} then \arg f=\frac{\theta-\phi}{2}+2k\pi. Now consider (i) at various points around the circle 1/2 e^{it}. At 1/2, \theta=0 and \phi=\pi so \arg f(1/2)=-\pi/2. Now start going around that circle and show how the term \frac{\theta-\phi}{2} changes smoothly between the values of -\pi/2 and 3\pi/2 except when you cross the line segment between zero and one and therefore the function f(z)=\sqrt{z/(z-1)} has a single-valued analytic component over this circle except for the line segment (branch-cut) between zero and one.
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