There are an increasing number of apparent correspondences between EM this semester and our section on complex analysis in math methods last semester. These are just notes on a few of them. Uniqueness of the Electrostatic Potential Solution and Liouville's theorem After stating that we would be solving Poisson's equation to determine electrostatic potentials, our professor then launched into a proof that the solutions, once found, would be unique. We first defined a potential psi equal to the difference between non-unique solutions, (assuming for the moment in our proof by contradiction that there could be more than one unique solution). We placed psi back into Poisson's equation and ran through the following steps: Ultimately we wound up proving that at best psi is a constant, but that it must be zero everywhere on the surface that defines the Dirichlet boundary conditions that the two 'non-unique' solutions both satisfy, so it's constant value ...