ABSTRACT
This chapter introduces the unifying concepts of eigenfunctions and eigenvalues of the Laplace operator, working primarily on rectangles. It considers the standard higher—dimensional heat, wave and Laplace's equations in terms of rectangular coordinates. The chapter utilizes multiple Fourier series and Fourier transforms to solve initial/boundary-value problems for these partial differential equation (PDE). It proves a uniform convergence theorem for eigenfunction expansions of suitable functions on a rectangle. The chapter deals with the standard PDEs written in terms of spherical coordinates. The Laplace operator on a sphere is defined in a geometrically natural way, and its eigenfunctions are introduced. The chapter also considers a number of special functions, such as Bessel functions and their use in solving heat problems in cylinders and in expressing the vibrational modes of a circular drum. It solves Schrodinger's equation for the quantum—mechanical description of the energy states of the electron in a hydrogen atom.
