It is known that the for the Cartesian Laplace equation (defined over $x\in[0,a],y\in[0,b], z\in[0,c]$) in three-dimensions with homogeneous Dirichlet conditions on both the $x$ and $y$ faces the form of the general solution is:
$$U(x,y,z)=\sum_{n,m=1}^{\infty}Z(z)\sin\bigg(\frac{n\pi x}{a}\bigg)\sin\bigg(\frac{m\pi y}{b}\bigg)$$
What will be the form of the general solution if both the faces on the $x$ and $y$ boundaries are homogeneous Neumann in nature ? Will the $x,y$ functions just convert to $\cos$ ?
If someone can provide the same result for the Robin conditions, that would be a certainly added advantage.