ABSTRACT
A Fourier series is a way of decomposing a function into a possibly infinite sum of “harmonic components”. We call those pieces “harmonic” because they are sinusoidal functions whose frequencies are multiples of a common base frequency. Those functions are also solutions of harmonic oscillator problems. We call them components because decomposing a function in this way is like decomposing a vector into orthogonal components, as we did in Section 2.4, specifically Corollary 2.7 in Section 2.4. Suppose f (x) is a function defined only on the interval [−L,L ]. Here, L is an unspecified
positive number. The Fourier series “expansion,” that is, decomposition into components, is given by
f (x) .= fs(x) = a02 + ∞∑
( an cos
(nπx L
) + bn sin
(nπx L
)) , (9.1)
where the real constants a0; a1, a2, a3, ...; b1, b2, b3, ... are called the Fourier coefficients. The notation f (x) .= fs(x) means that f is “represented” by fs, the Fourier series for f . We
will see that fs(x) may be unequal to f (x) at some value(s) of x; even worse, the Fourier series fs(x) may fail to converge at some value(s) of x. So, in principle, fs(x) and f (x) may be different functions. Note that even though the original function f (x) is defined only for −L ≤ x ≤ L, its
Fourier series fs(x) may be defined for all x and is periodic with period 2L, that is, fs(x+ k · 2L) = fs(x) for all integers k. One way to think of the relationship of fs to f is that it is like the relationship of your
computer game’s “avatar” to you. In some circumstances your avatar may behave just like you, but perhaps not always. Another way to think of the relationship is between a movie’s “stunt double” and the actress she replaces: From certain viewpoints they may look and behave exactly alike. In fact, the actress may do the stunt herself so she is being her own stunt double.
