I am trying to solve Laplace equation in cartesian coordinates, on a rectangle defined by $x_1<x<x_2$ and $-y_0<y<y_0$: $$\nabla^2 g=0$$ with $$g(x,y=\pm y_0)=f(x)\\ g(x_1,y)=f(x_1) \\ \frac{\partial g}{\partial x}(x_2,y)=f'(x_2) $$ where $f(x)=e^{2x}$.
Using separation of variables we can quickly find that the solution should be a (continuous) sum of the basis functions $$(x,y)\rightarrow \left(a(\lambda)e^{\sqrt \lambda x}+b(\lambda)e^{-\sqrt \lambda x}\right)\left(c(\lambda)e^{i\sqrt \lambda y}+d(\lambda)e^{-i\sqrt \lambda x}\right)$$ where $a, b,c,d$ are the coefficient functions to be determined. $\lambda$ is a real number.
Usually there's some symmetry, constraints, or nice properties of the Fourier transform one can use to find the coefficients. Here I don't know what to do. $f$ is not harmonic but it's still an exponential, so it makes me hope there might still be a way to derive something, but I haven't been able to get anything.