Fourier Series Expansion

Abstract

In Chap. 1, it was shown, mostly by using graphics, that various waves can be expressed by a summation of sine and cosine functions, i.e., by the Fourier series (see Eq. 1.5). In this chapter, first, a method of determining coefficients of Fourier series will be given. A key idea is the integral of the products of sine and cosine functions. It was shown that an addition of sine and cosine functions with the same frequency can be combined into one sine or cosine function by introducing a phase term (see Eq. (1.15)). It is also possible to express an arbitrary function by a combination of even and odd functions. The former and the latter can be expressed by cosine and sine functions, respectively. The next step is the expression of a Fourier series by complex exponential functions. The coefficients in this case are also complex, but since the mathematical manipulations are simpler, this method will be used most of the time hereafter.