the first-order monogenic equation: $\Box\Psi=0$ \[or the generalized Dirac eq. (3.2a)\] more directly lends itself to integral solution than the associated second-order harmonic equation \[or the generalized Klein-Gordon eq. (3.1)\]. For example, time independent (i.e. ) “static” electromagnetism can be easily re-expressed in Cauchy integral form, even when non-homogeneous source terms are included. A typical electrostatic example would be to calculate the electric field $\vector{E}$ at any point inside a region knowning the field on the boundry (equivalently knowing the surface charge density $d\sigma=
\vector{E}\cdot d\vector{A}$) and the distribution of (scalar) charge $\rho=\nabla \vector{E}$ inside the region[@Jackson]. On the other hand, the Bergman kernel would only provide a method of calculating the electric field at a point in the region, if one already knows the complete solution everywhere inside the region. I am not aware of any classical problem which makes use of this circular relationship, however it possibly could be used to generate a perturbation series approximation.

There are additional constraints imposed by fundamental physical laws which further limit the usefulness of integral formulations in describing tangible phenomena. For example, consider the : $(\nabla^2+\lambda^2)\phi(\vector{X})=0$, of which integral solutions to the factored meta-monogenic form have been recently treated by Shapiro[@Shapiro]. 