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Normal derivative at boundary Laplace's equation half plane

  1. May 1, 2014 #1

    Given a holomorphic function [itex] u(x,y) [/itex] defined in the half plane ([itex] x\in (-\infty,\infty), y\in (-\infty,0)[/itex]), with boundary value [itex] u(x,0) = f(x) [/itex], the solution to this equation (known as the Poisson integral formula) is

    [tex] u(x,y) = \int_{-\infty}^{\infty} \frac{y\ f(t) }{(t-x)^2 +y^2} \ dt [/tex].

    I have two questions. First, how does one prove

    [itex] u(x,0) = f(x) [/itex],

    and related, but what I'm more interested in, how do I find

    [itex] \left.\frac{\partial u(x,y) }{\partial y}\right|_{y=0} [/itex]

    The subtleties on how take these limits (namely [itex] y\to 0; \ y<0[/itex]) appropriately is very much beyond me. Any suggestions/references on how to answer this would be very appreciated.


  2. jcsd
  3. May 4, 2014 #2
    I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
  4. May 7, 2014 #3
    I'm not entirely sure about your first question. You defined the upper half-plane problem to have the BV of f(x). I don't think you would need to prove that, because that is just a definition in the problem. Correct me if I'm wrong.
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