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!

Complex analysis (conformal?) mapping question probably easy

  1. Sep 17, 2009 #1
    1. The problem statement, all variables and given/known data
    We're supposed to find a bijective mapping from the open unit disk [itex]\{z : |z| < 1\}[/itex] to the sector [itex]\{z: z = re^{i \theta}, r > 0, -\pi/4 < \theta < \pi/4 \}[/itex].


    2. Relevant equations



    3. The attempt at a solution
    This is confusing me. I tried to find a function that would map [itex][0,1)[/itex], which is the set of possible values of [itex]r[/itex] in the domain, injectively onto [itex](0,\infty)[/itex], which is the set of possible values of [itex]r[/itex] in the range. The best thing I could come up with is [itex]f(r) = \dfrac{1}{r(1-r)} - 4[/itex], but this is clearly not one-to-one, and it hits zero. What's more, I'm not sure how to find a function that will map the possible values for [itex]\text{Arg }z[/itex], which are [itex]-\pi < \text{Arg }z \leq \pi[/itex], injectively onto [itex](-\pi/4, \pi/4)[/itex].
     
  2. jcsd
  3. Sep 17, 2009 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi AxiomOfChoice! :smile:
    why?? :redface:

    Hint: get the boundary right, and everything else should fit in. :wink:
     
  4. Sep 17, 2009 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Don't try and mess around individually with r and theta. Just think about analytic functions. For example 1/(1-z) maps the disk into a half plane, right? Now find another function that can take a half plane into a wedge. Put them together.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Complex analysis (conformal?) mapping question probably easy
Loading...