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13.2 The Fourier Series of a Function 433

y

x

FIGURE 13.3 Convergence of a Fourier
series at a jump discontinuity.

At both −L and L this Fourier series converges to

1
( f (L−) + f (−L+)).
2
At any point in (−L, L) at which f (x) is continuous, the Fourier series converges to f (x),
because then the right and left limits at x are both equal to f (x). At a point interior to the
interval where f has a jump discontinuity, the Fourier series converges to the average of the left
and right limits there. This is the point midway in the gap of the graph at the jump discontinuity
(Figure 13.3). The Fourier series has the same sum at both ends of the interval.

EXAMPLE 13.4
2
Let f (x)= x − x for −π ≤ x ≤π. In Example 13.1 we found the Fourier series of f on [−π,π].
Now we can examine the relationship between this series and f (x).

f (x) = 1 − 2x is continuous for all x, hence f is piecewise smooth on [−π,π].For −π<
2
x <π, the Fourier series converges to x − x .Atboth π and −π, the Fourier series converges to
1 1
2
2
( f (π−) + f (−π+)) = ((π − π ) + (−π − (−π) ))
2 2
1
2
2
= (−2π ) =−π .
2
Figures 13.4, 13.5, and 13.6 show the fifth, tenth and twentieth partial sums of this Fourier
series, together with a graph of f for comparison. The partial sums are seen to approach the
function as more terms are included.
EXAMPLE 13.5
x
Let f (x) = e . The Fourier coefficients of f on [−1,1] are
1

x
−1
a 0 = e dx = e − e ,
−1
1 −1 n
x (e − e )(−1)
a n = e cos(nπx)dx = ,
2
−1 1 + n π 2

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October 14, 2010 14:57 THM/NEIL Page-433 27410_13_ch13_p425-464