This is concerning Poisson's equation with oblique boundary condition (Gilbarg Trudinger p121)
We let $\Gamma(|x-y|)$ denote the fundamental solution to Laplace's equation. Also, let $x-y^{*} = (x_1-y_1, \cdots ,x_{n-1}-y_{n-1},x_n+y_n)$. Finally, let $\zeta = \frac{(x-y^{*})}{|x-y^{*}|}$
I don't understand the computations to get from this
$ \Theta = -2b_n\int_0^{\infty}{e^{as}D_n\Gamma(x-y^{*}+\textbf{b}s)ds}$ where $a\leq 0$
to this
$ \Theta = -|x-y^{*}|^{2-n}\left( (\frac{2 b_n}{n\omega_n})\int_0^{\infty}e^{a|x-y^{*}|s}\frac{\zeta_n+b_ns}{(1+2({\textbf{$\zeta$}\cdot\textbf{b})s+s^2)^{\frac{n}{2}}}}ds\right)$
where $\omega_n$ is the volume of the n-ball. I guess I'm getting stuck in one regard because we only define the fundamental solution for $|x-y|$, and I've never seen something where you are adding a vector inside. Also, how do we get the additional term in the exponent? If someone could point me in the right direction, I would appreciate it. Thanks.