Mathematical Methods for Physics and Engineering - Matematica.NET

12Fourier seriesWe have already discussed, in chapter 4, how complicated functions may beexpressed as power series. However, this is not the only way in which a functionmay be represented as a series, and the subject of this chapter is the expressionof functions as a sum of sine and cosine terms. Such a representation is called aFourier series. Unlike Taylor series, a Fourier series can describe functions that arenot everywhere continuous and/or differentiable. There are also other advantagesin using trigonometric terms. They are easy to differentiate and integrate, theirmoduli are easily taken and each term contains only one characteristic frequency.This last point is important because, as we shall see later, Fourier series are oftenused to represent the response of a system to a periodic input, and this responseoften depends directly on the frequency content of the input. Fourier series areused in a wide variety of such physical situations, including the vibrations of afinite string, the scattering of light by a diffraction grating and the transmissionof an input signal by an electronic circuit.12.1 The Dirichlet conditionsWe have already mentioned that Fourier series may be used to represent somefunctions for which a Taylor series expansion is not possible. The particularconditions that a function f(x) must fulfil in order that it may be expanded as aFourier series are known as the Dirichlet conditions, and may be summarised bythe following four points:(i) the function must be periodic;(ii) it must be single-valued and continuous, except possibly at a finite numberof finite discontinuities;(iii) it must have only a finite number of maxima and minima within oneperiod;(iv) the integral over one period of |f(x)| must converge.415