Notice that in the Fourier series of the square wave ( 4.23 ) all coefficients vanish, the series only contains sines. This is a very general phenomenon for so-called even and odd functions.
These have somewhat different properties than the even and odd numbers:
Question: Which of the following
functions is even or odd?
a)
, b)
, c)
, d)
, e)
, f)
Answer: even: d, f; odd: a, b, c, e.
Now if we look at a Fourier series, the Fourier cosine series
(4.28) |
describes an even function (why?), and the Fourier sine series
(4.29) |
an odd function. These series are interesting by themselves, but play an especially important rôle for functions defined on half the Fourier interval, i.e., on instead of . There are three possible ways to define a Fourier series in this way, see Fig. 4.2
Of course these all lead to different Fourier series, that represent the same function on . The usefulness of even and odd Fourier series is related to the imposition of boundary conditions. A Fourier cosine series has at , and the Fourier sine series has . Let me check the first of these statements:
(4.30) |
As an example look at the function , , with an even continuation on the interval . We find
So, changing variables by defining so that in a sum over all runs over all odd numbers,
(4.32) |