In Chapter 6 we discussed how complicated functions f(x) may be expressed as power series. Although they were not presented as such, the power series could all be viewed as linear superpositions of the monomial basic set of functions, namely 1, x, x
2, x
3, … xn
, … Natural though this set may seem, they are in many ways far from ideal: for example they possess no mutual orthogonality properties, a characteristic that is generally of great value when it comes to determining, for any particular function, the multiplying constant for each basic function in the sum. Moreover, this particular set of basic functions can only be used to represent continuous functions.
In the case of original functions f(t) that are periodic, some improvement on this situation can be made by using, as the basic set, sine and cosine functions. For a function with period T, say, the set of sine and cosine functions with arguments 2πnt/T, for all n ≥ 0, form a suitable basic set for expressing f as a series; such a representation is called a Fourier series. One great advantage they possess over the monomial functions is that they are mutually orthogonal when integrated over any continuous period of length T, i.e. the integral from t
0 to t
0 + T of the product of any sine and any cosine, or of two sines or cosines with different values of n, is equal to zero.