Page 471 - Advanced engineering mathematics

13.4 Integration and Differentiation of Fourier Series 451

Proof Begin with the fact that, from the Fourier convergence theorem,

∞
1
f (x) = a 0 + (a n cos(nπx/L) + b n sin(nπx/L)).
2
n=1
Multiply this series by f (x) to get

∞
1
2
f (x) = a 0 f (x) + (a n f (x)cos(nπx/L) + b n f (x)sin(nπx/L)).
2
n=1
We can integrate this equation term by term (Theorem 13.4). In doing this, observe that the
integrals in the series on the right are Fourier coefficients. This yields Parseval’s theorem.

EXAMPLE 13.15
We will apply Parseval’s theorem to sum a series. The Fourier coefficients of cos(x/2) on [−π,π]
are
1 π 4
a 0 = cos(x/2)dx =
π −π π

and, for n = 1,2,··· ,,

1 π 4 (−1) n
a n = cos(x/2)cos(nx)dx =− .
2
π −π π 4n − 1
Each b n = 0 because cos(x/2) is an even function. By Parseval’s theorem,

2 ∞ n 2 π
1 4 4 (−1) 1
2
+ = cos (x/2)dx = 1.
2
2 π π 4n − 1 π
n=1 −π
After some routine manipulation, this yields
∞ 2
1 π − 8

= .
2
(4n − 1) 2 16
n=1
We conclude this section with sufficient conditions for a Fourier series to converge
uniformly.

THEOREM 13.8

Let f be continuous on [−L, L], and let f be piecewise continuous. Suppose f (L) = f (−L).

Then the Fourier series for f (x) on [−L, L] converges absolutely and uniformly to f (x) on
[−L, L].
A proof is outlined in Problem 6.

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October 14, 2010 14:57 THM/NEIL Page-451 27410_13_ch13_p425-464