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Math Help - Branch points

  1. #1
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    Branch points

    I am trying to find the branch points of w = Log(z(z-1)) and I know that Log is undefined for z = 0 and 1.

    Let theta = arg(z) and phi = arg(z-1)

    Log(z(z-1)) = ln[abs(z)abs(z-1)e^(itheta)e^(iphi)] + iArg[abs(z)abs(z-1)e^(itheta)e^(iphi)]

    So, if I traverse a closed path abs(z) and abs(z-1) remain the same. And theta and phi are multiplied by an integer (1).

    If I go around the points z = 0 and z = 1 (x = 1, y = 0), then obviously theta and phi both change from some angle to that angle + 2pi, so the value of w will be the same around closed path.

    If I go in a loop between z = 0 and z = 1, starting at theta = 0, theta ranges from 0 to 2 pi and phi is 0 at start and finish. Once again, this just yields same value of w.

    I can't think of any other paths, so I can't decide if there are or are not branch points because I can't find any paths for which the values of w disagree.
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  2. #2
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    Quote Originally Posted by amoeba View Post
    I am trying to find the branch points of w = Log(z(z-1)) and I know that Log is undefined for z = 0 and 1.

    Let theta = arg(z) and phi = arg(z-1)

    Log(z(z-1)) = ln[abs(z)abs(z-1)e^(itheta)e^(iphi)] + iArg[abs(z)abs(z-1)e^(itheta)e^(iphi)]

    So, if I traverse a closed path abs(z) and abs(z-1) remain the same. And theta and phi are multiplied by an integer (1).

    If I go around the points z = 0 and z = 1 (x = 1, y = 0), then obviously theta and phi both change from some angle to that angle + 2pi, so the value of w will be the same around closed path.

    If I go in a loop between z = 0 and z = 1, starting at theta = 0, theta ranges from 0 to 2 pi and phi is 0 at start and finish. Once again, this just yields same value of w.

    I can't think of any other paths, so I can't decide if there are or are not branch points because I can't find any paths for which the values of w disagree.
    The branch points of w = \text{Log} (z(z-1)) are z = 0 and z = 1. A branch transition does not occur when a complete circuit is made about both branch points. It's only when the circuit encloses a single branch point that a branch transition occurs.
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