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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.

Thanks!

Nick

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# Normal derivative at boundary Laplace's equation half plane

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