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Math Help - Help with branch cuts

  1. #1
    Senior Member Danneedshelp's Avatar
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    Help with branch cuts

    Q: How do I algebraically find the appropriate branch cut of a function? For example, lets consider f(z)=log(z^{2}).

    If we choose to use the principle branch of the log function then f'(z)=\frac{2}{z} which is undefined at 0. I know that much, but the text I am using says log(z^{2}) is analytic on the set A=\{z:z\neq\\0 and arg(z)\neq\\+\frac{\pi}{2} or  -\frac{\pi}{2}\}. I am not seeing why it is this set.

    In general, is there a way to assume nothing about your function other than maybe the log is defined on the principle branch and solve for the appropriate branch cuts? I can find the branch points, but not sure how to construct the proper cuts.

    Thanks
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  2. #2
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    Quote Originally Posted by Danneedshelp View Post
    Q: How do I algebraically find the appropriate branch cut of a function? For example, lets consider f(z)=log(z^{2}).

    If we choose to use the principle branch of the log function then f'(z)=\frac{2}{z} which is undefined at 0. I know that much, but the text I am using says log(z^{2}) is analytic on the set A=\{z:z\neq\\0 and arg(z)\neq\\+\frac{\pi}{2} or  -\frac{\pi}{2}\}. I am not seeing why it is this set.

    In general, is there a way to assume nothing about your function other than maybe the log is defined on the principle branch and solve for the appropriate branch cuts? I can find the branch points, but not sure how to construct the proper cuts.

    Thanks
    Although convention might dictate a particular branch cut, in reality many branch cuts are possible. To define a branch cut, you first need to know where the branch point(s) is(are).
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