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Homework Help: Analytic region

  1. Mar 13, 2012 #1
    1. The problem statement, all variables and given/known data

    Describe the largest region in which g(z) = 1/[Log(z)-(i*pi)/2] is analytic?

    3. The attempt at a solution

    Analytic is where the cauchy-euler equations hold, so I tried to take the partials to and set them equal so I could define a domain. I am not sure if the partials are correct.

    I also tried to look at it from the following point of view. It would be analytic where the domain of g(z) is valid. My line of thinking is Log(z) is the principal log and it has a branch cut at pi. Thus pi is not in the domain. Also 1/z where z≠0 so Log(z)-(i*pi)/2 ≠ 0.
    Log(z) ≠ (i*pi)/2. I think Log(i) = (i*pi)/2. If so then the parts not in the domain are pi and i.

    Does either attempt sound like a possible solution?
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Mar 13, 2012 #2
    branch point of Log[z] is 0, the branch cut can be arbitrarily chosen to go from 0 to infinity, so if you choose arg(z) in (pi/2,5pi/2) then g(x) is analytic everywhere except positive imag axis
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