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436 CHAPTER 13 Fourier Series

13.2.1 Even and Odd Functions

A function f is even on [−L, L] if its graph on [−L,0] is the reflection across the vertical
axis of the graph on [0, L]. This happens when f (−x) = f (x) for 0 < x ≤ L. For example,
x 2n and cos(nπx/L) areevenonany [−L, L] for any positive integer n.
A function f is odd on [−L, L] if its graph on [−L,0) is the reflection through the
origin of the graph on (0, L]. This means that f is odd when f (−x)=− f (x) for 0<x ≤ L.
For example, x 2n+1 and sin(nπx/L) are odd on [−L, L] for any positive integer n.

Figures 13.8 and 13.9 show typical even and odd functions, respectively.
A product of two even functions is even, a product of two odd functions is even, and a
product of an odd function with an even function is odd.
If f is even on [−L, L] then
L L
f (x)dx = 2 f (x)dx
−L 0
and if f is odd on [−L, L] then
L
f (x)dx = 0.
−L
These facts are sometimes useful in computing Fourier coefficients. If f is even, only the cosine
terms and possibly the constant term will appear in the Fourier series, because f (x)sin(nπx/L)
is odd and the integrals defining the sine coefficients will be zero. If the function is odd
then f (x)cos(nπx/L) is odd and the Fourier series will contain only the sine terms, since
the integrals defining the constant term and the coefficients of the cosine terms will be
zero.

35
60
30
25 40

20 20
x
15 –4 –2 0 0 2 4
10
–20
5
–40
0
–6 – 4 – 2 0 2 4 6
x –60
FIGURE 13.8 A typical even function. FIGURE 13.9 A typical odd function.

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October 14, 2010 14:57 THM/NEIL Page-436 27410_13_ch13_p425-464