Mathieu functions and numerical solutions of the Mathieu equation

We review the full spectrum of solutions to the Mathieu differential equation y'' + [a - 2q cos(2z)]y = 0, and we describe a numerical algorithm which allows a flexible approach to the computation of all the Mathieu functions. We use an elegant and compact matrix notation which can be readily implemented on any computing platform. We give some explicit examples (written in the programming language Scilab) that provide a ready-to-use package for solving the Mathieu differential equation and related applications in several fields.