Background: I have obtained the solution to the Laplace equation in the upper half-plane $(y \geq 0 , -\infty < x < \infty)$ with a function value of zero and prescribed normal derivative of $\partial_y f = g(x)$ at the boundary $$f(x,y) = \frac{1}{2 i} \int_{x-iy}^{x+iy} g(\xi) \; \mathrm{d}\xi \, . $$ My goal is to map this solution in the half-plane to the quadrant $(x,y \geq 0)$ with boundary conditions of $\partial_y f = g(x)$ at $y=0$ and $\partial_x f = h(y)$ at $x=0$ and $f=0$ along the boundary. I have asked this question in another post and have received advice. However, I have not been able to figure out the mapping process and how to impose the boundary conditions.
My approach: Using the map $f(z) = \sqrt{z}$ to map the half-plane to the quadrant gives $$f(x,y) = \frac{1}{2 i} \int_{\sqrt{x^2-y^2-i2xy}}^{\sqrt{x^2-y^2+i2xy}} g(\xi) \; \mathrm{d}\xi \; . $$ Function $f(x,y)$ satisfies the Laplace equation and has a value of zero at $x=0$ or $y=0$. It has derivatives $$ \partial_y f = g(x) \hspace{4mm} \text{at} \hspace{4mm} y=0 \\ \partial_y f = g(i y) \hspace{4mm} \text{at} \hspace{4mm} x=0 \; . $$ So it seems that I cannot prescribe two independent normal derivative boundary conditions. Can you tell me where my approach is wrong?