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Homework Help: The Poisson Kernel

  1. Dec 27, 2011 #1

    A_B

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    1. The problem statement, all variables and given/known data
    Can you look at Poisson's formula for a half plane as a limit case of Poisson's formula for a disk?
    http://en.wikipedia.org/wiki/Poisson_kernel

    I can find lots of information about the Poisson kernel for a disk, but not for the half plane. I do know on can mat the unit disk to the half plane using a Möbius transformation, but that's not what this question is asking (that comes in a later question, and is easily looked up in literature).

    I can't give an "attempt at a solution" because what' I've tried hasn't produced anything useful. I tried interpreting the formula for the disk in polar coordinates, then transforming to Cartesian and looking for a suitable limit, to no avail.

    This is for a project given for a first course Differential Equations.

    Thanks in advance
    A_B
     
  2. jcsd
  3. Dec 30, 2011 #2

    A_B

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    Having worked a little more on the project, I also got to know the Poisson formulas better, and it seems to me like it doesn't make any sense at all to look at the Poisson formula for the half-plane as a limit of the Poisson formula for the disk. They are transformed into each other by a conformal map, not at all by taking some limit.
    But surely the professor wouldn't put the question in a project if it can't be done?!

    A_B
     
  4. Jan 8, 2012 #3

    BVM

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    I'm stuck on exactly the same problem.

    My initial aproach was to define the poisson kernel on a circle with radius r, and then take the limit of the poisson kernel with [itex]r \rightarrow ∞[/itex], but I kept on getting 0 as result.

    I have found, however, that we can rewrite [tex]\frac{1 - r^2}{1 - 2rcos(2πt) + r^2}[/tex] as [tex]\frac{1 - z\bar{z}}{1 - z - \bar{z} + z\bar{z}}[/tex] with [tex]z =r e^{2πiθ}.[/tex]

    If that would be of any help to you...
     
    Last edited: Jan 8, 2012
  5. Jan 11, 2012 #4
    We seem to have pretty much the same project. Do your projects also include Schwarz–Christoffel mapping and möbiustransformation?
    Did you find the solution to the limit problem yet?

    I don't really see why you were taking the limit of r? My approach was rather to write the formular in terms of an arbitrary radius R and then taking the limit R -> infinity though I still don't see how this can solve the dirichlet-problem.

    regards
     
  6. Jan 11, 2012 #5

    A_B

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    Whaha! It's almost a convention here! I'm at KUL as well, so is BVM.
    I haven't cracked it yet, but on facebook someone said she solved it starting from the Poisson kernel for the disk, but applying it to a disk of radius R with center (o, R). And then letting R go to infinity. I haven't attempted to repeat this approach yet, doing that tomorrow.

    If any of you find something interesting, please let me know!
     
  7. Jan 11, 2012 #6
    Did you solve the rest of it? And any idea of how thorough they expect us to be by solving those questions? There are quite a lot of terms en definitions that were not encountered in the lectures, though are necessary for solving those problems.

    But yes translating the centre of the circle seems like a good approach.

    good luck!
     
  8. Jan 11, 2012 #7
    "en" oops
     
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