Abstract

This chapter, which is totally independent of the remaining parts of this monograph, studies the fact that the solutions of the scalar Helmholtz equation or the vectorial Maxwell system in balls can be expanded into certain special “wave functions.” We begin by expressing the Laplacian in spherical coordinates and search for solutions of the scalar Laplace equation or Helmholtz equation by separation of the (spherical) variables. It will turn out that the spherical parts are eigensolutions of the Laplace- Beltrami operator while the radial part solves an equation of Euler type for the Laplace equation and the spherical Bessel differential equation for the case of the Helmholtz equation.

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