Green's function for $-\Delta$ on the plane is $-\frac{1}{2\pi}\ln\lVert x-y \rVert$
a) Using the method of reflection, compute the green's function for Laplace's equation $-\Delta u=0$
b) Compute the green's function for the quadrant with the following other boundary condition Neuwann on the x-axis and dirichlet on the y-axis
My answers
a)$U(x)= V\lvert x\rvert$ where, $x=(x,x^{2},x^{3},...x^{n})$
$-\Delta u = -\frac{\partial^{2}u}{\partial x^{2}} - \frac{\partial^{2}u}{\partial y^{2}}=0$
taking derivative of $U$
$Ux_i = V'\frac{x_i}{\lvert x\rvert}$
$Ux_ix_i = V''\frac{x_i^{2}}{\lvert x\rvert^{2}}$ + $V'\frac{\lvert x\rvert -\frac{x_i^{2}}{\lvert x\rvert}}{\lvert x\rvert^{2}}$
therefore
$-\Delta u= \sum Ux_ix_i= \sum V''\frac{x_i^{2}}{\lvert x\rvert^{2}} + V'\frac{\lvert x\rvert-\frac{x_i^{2}}{\lvert x\rvert}}{\lvert x\rvert^{2}} $
I'm stuck here.