9 - The Fourier Series

Summary

In this chapter, we derive Sturm–Liouville theory that introduces a broad class of eigenfunctions that are convenient to use for representing functions. Sturm–Liouville theory provides the basis of the Fourier-series method of representing functions that is the main focus of the chapter and that also is the foundation of Fourier analysis. We show how to calculate Fourier series and to use Fourier series to obtain the solution of boundary-value problems posed in Cartesian coordinates. It is seen that the main advantage of an eigenfunction approach for solving boundary-value problems is that either the inhomogeneous source term in the differential equation or the boundary values may be time dependent, which they cannot be in the method of separation of variables.