ABSTRACT
Suppose, however, that we are dealing with a function F(x) that is not periodic in x. We can think of such a function as one that is periodic in x with a period L that tends to infinity. Does this mean that we can still represent F(x) as a Fourier series? Consider what happens to the series (8.2) in the limit L → ∞ or, equivalently, δk → 0. The series is basically a weighted sum of sinusoidal functions whose wavenumbers take the quantized values kn = n δk. Moreover, as δk → 0, these values become more and more closely spaced. In fact, we can write
F(x) = ∑
Cn δk
cos(n δk x) δk + ∑
S n δk
sin(n δk x) δk. (8.9)
In the continuum limit, δk → 0, the summations in the previous expression become integrals, and we obtain
F(x) = ∫ ∞
−∞ C(k) cos(k x) dk +
∫ ∞
−∞ S (k) sin(k x) dk, (8.10)
where k = n δk, C(k) = C(−k) = Cn/(2 δk), and S (k) = −S (−k) = S n/(2 δk). Thus, for the case of an aperiodic function, the Fourier series (8.2) morphs into the socalled Fourier transform (8.10). This transform can be inverted using the continuum limits (i.e., the limit δk → 0) of Equations (8.7) and (8.8), which are readily shown to be
C(k) = 1
2π
∫ ∞
−∞ F(x) cos(k x) dx,
S (k) = 1
2π
∫ ∞
−∞ F(x) sin(k x) dx,
(8.11)
(8.12)
respectively. (See Exercise 8.5.) The previous equations confirm that C(−k) = C(k) and S (−k) = −S (k). The Fourier-space (i.e., k-space) functions C(k) and S (k) are known as the cosine Fourier transform and the sine Fourier transform of the realspace (i.e., x-space) function F(x), respectively. Furthermore, because we already know that any periodic function can be represented as a Fourier series, it seems plausible that any aperiodic function can be represented as a Fourier transform. This is indeed the case. Equations (8.10)–(8.12) effectively enable us to represent a general function as a linear superposition of sine and cosine functions. Let us consider some examples.
