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432 CHAPTER 13 Fourier Series

2

1
x
–4 –2 0 0 2 4 6

–1

–2

–3
FIGURE 13.2 f in Example 13.2.

EXAMPLE 13.2
Let

x for −3 ≤ x < 2,
f (x) =
1/x for 2 ≤ x ≤ 4.
f is piecewise continuous on [−3,4], having a single discontinuity at x = 2. Furthermore,
f (2−) = 2 and f (2+) = 1/2. A graph of f is shown in Figure 13.2.

f is piecewise smooth on [a,b] if f is piecewise continuous and f exists and is continuous

at all but perhaps finitely many points of (a,b).

EXAMPLE 13.3
The function f of Example 13.2 is differentiable on (−3,4) except at x = 2:

1 for −3 < x < 2

f (x) =
−1/x 2 for 2 < x < 4.
This derivative is itself piecewise continuous. Therefore f is piecewise smooth on [−3,4].

We can now state a convergence theorem.

THEOREM 13.1 Convergence of Fourier Series

Let f be piecewise smooth on [−L, L]. Then, for each x in (−L, L), the Fourier series of f on
[−L, L] converges to
1
( f (x+) + f (x−)).
2

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October 14, 2010 14:57 THM/NEIL Page-432 27410_13_ch13_p425-464