our model must be able to represent the fields of arbitrary physical magnets like those in Figures 1 and 2, which looking to the CAD models, have completely different boundary conditions.

If we were to stop here and write down every possible solution to our PDE, our model would be far too complicated to be useful. We have that $\lambda_x$ and $\lambda_z$ may each take on three different signs (zero being a sign). For each possible sign, $\lambda_x$ and $\lambda_z$ yield two linearly independent solutions to whichever ordinary differential equation (ODE) they are involved in. Considering only these cases, there are already 24 different possible forms which the vector potential basis functions may take on. That number becomes even larger when we account for the fact that in addition to the signs of $\lambda_x$ and $\lambda_z$, the sign of $\lambda_y$ also depends on their relative magnitudes. Fortunately, with further insight and by adding a relatively small number of extra parameters we may vastly simplify the model. First, since sine’s and cosine’s yield a complete basis for scalar valued functions, without loss of generality we may assert that $\lambda_z \equiv -k_z^2 < 0$, which forces $J(z)$ and $M(z)$ to be (co)sinusoidal. 