# Normal derivative at boundary Laplace's equation half plane

1. May 1, 2014

### nickthequick

Hi,

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

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

I have two questions. First, how does one prove

$u(x,0) = f(x)$,

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

$\left.\frac{\partial u(x,y) }{\partial y}\right|_{y=0}$

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

Thanks!

Nick

2. May 4, 2014