Those solutions to this equation, the simplest of which is one-half times the square root of one over pi, are eigenfunctions of the square of the orbital angular momentum operator.
Solutions of this equation that map to Euclidean space correspond to instantons with action given by the Dirichlet energy. For a certain set of solutions to this equation, taking the complex conjugate is equivalent to negating the second degree parameter and multiplying by the Condon-Shortley phase. Those solutions to this equation, the simplest of which is one-half times the square root of one over pi, are eigenfunctions of the square of the orbital angular momentum operator. Substituting this equation into the expression for the energy of a dipole in a constant magnetic field results in Earnshaw's theorem. In two dimensions, this equation's fundamental solution is "negative log r over two pi," and in three dimensions, it is the electric potential generated by a point charge. This equation governs the velocity potential of an incompressible flow and the steady-state distribution of heat. For 10 points, harmonic functions satisfy what equation written "del squared u equals zero"?
Solutions of this equation that map to Euclidean space correspond to instantons with action given by the Dirichlet energy.
Laplace's equation [do not accept or prompt on more general answers such as "Poisson's equation" or "heat equation"]