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I Branch cut

  1. Jun 8, 2017 #1
    <Moderator note: Merger of two threads on the topic.>

    Hello! I am reading some basic stuff on complex integrals using branch cuts and i found the problem in the attachment. I am not sure I understand why the branch cut is along ##R^+##. I thought that branch cut is, loosely speaking, a line where the function is not continuous (and thus not holomorphic). But in the presented problem, the function is continuous on ##R^+## as ##lim_{\theta \to 0} = \sqrt{r}## and ##lim_{\theta \to 2\pi} = -\sqrt{r}##. The limits are not equal, but they don't have to be, as the funtion is not defined for ##\theta = 2\pi##. However, the function is not continuous for ##\theta = \pi##, as, coming from above and below x-axis, gives different values for ##sin(\theta)##. So, isn't the branch cut on ##R^-##, or did I get something wrong about the definition of branch cut? Thank you!
     
    Last edited by a moderator: Jun 8, 2017
  2. jcsd
  3. Jun 8, 2017 #2
    Hello! I understand that the branch cuts are meant to prevent a complex function from being multivalued. So, as any complex number ##z## can be written in different ways (##z=\|z\|e^{i\theta}=\|z\|e^{i(\theta +2\pi)}## etc.), does this mean that any complex function has a branch cut, depending on the interval on which we define ##\theta##, (##[0,2\pi)## or ##[-\pi,\pi)##, etc)?
     
  4. Jun 8, 2017 #3
    Look, functions like ##f(z)=z^2##, gives you the same value for a particular ##z##, no matter you write ##z=||z||e^{i\theta}## or ##z=||z||e^{i(\theta+2\pi)}##.
    The problem arises when you deal with functions like ##g(z) = log (z)## or ##g(z) = z^{1/2}##. In those cases, you notice, ##z=||z||e^{i\theta}## or ##z=||z||e^{i(\theta+2\pi)}## give different values of ##g(z)##.
     
  5. Jun 8, 2017 #4

    mathwonk

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    the fact that theta is multivalued is irrelevant to the function e^(i.theta) since the periodicity of the exponential function cancels out the multivaluedness of theta.
     
  6. Jun 9, 2017 #5

    WWGD

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    Another perspective is that of branch points . You want a branch cut designed so that curves do not wind around the branch point. EDIT: For example, for logz, the Complex log, zero is the branch point and branch cuts disallow winding around it.
     
  7. Jun 10, 2017 #6
    But how do you calculate a branch cut in general? For example ##f(z)=\sqrt{z(z-1)}##, how do I get branch points and how do I get the number of values you get while going around the branch points? Thank you!
     
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