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Homework Help: Principal branch of the logarithm

  1. Jan 22, 2012 #1
    1. The problem statement, all variables and given/known data

    Define an analytic branch f(z) of w, such that f(z)=0 for the limit of z->[itex]\infty[/itex]

    Now what is f(1)?

    2. Relevant equations

    [itex]w=\frac{z+i}{z-i}[/itex]

    3. The attempt at a solution

    The branch cut of the logarithm is: [itex](-\infty,0)[/itex]
    All branches of the logarithm are:
    f(z)=Log(z)+iArg(z)=Log(z)+2i[itex]\pi[/itex]k

    But then f(1)=0, which is wrong.
     
    Last edited: Jan 22, 2012
  2. jcsd
  3. Jan 23, 2012 #2
    There was a typo in my first post:

    Of course this should be [itex]w=log(\frac{z+i}{z-i})[/itex]

    Anybody who understands it now?
     
  4. Jan 23, 2012 #3
    A guess is that the line segment of z=(i,-i) is mapped onto the normal branch cut of the logarithm (-inf,0). Therefore, f(1)=exp(iPi/2) because this is where i is located in the complex plane.
     
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