1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook