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512 CHAPTER 15 Special Functions and Eigenfunction Expansions

THEOREM 15.2 Convergence of Eigenfunction Expansions

Suppose ϕ n ,for n = 1,2,···, are the eigenfunctions of a Sturm-Liouville problem on [a,b].Let
f be piecewise smooth on [a,b] and let c n be given by equation (15.4). Then, for a < x < b,the
eigenfunction expansion ∞ c n ϕ n (x) converges to
n=1
1
( f (x+) + f (x−)).
2
In particular, if f is continuous at x, then this eigenfunction expansion converges to f (x).

It should not be surprising that this conclusion has the same form as that for convergent
Fourier series, since Fourier series are eigenfunction expansions.

EXAMPLE 15.3
The regular Sturm-Liouville problem

y + λy = 0; y (0) = y (π/2) = 0
2
has eigenvalues λ n =4n and eigenfunctions ϕ n (x)=cos(2nx) for n =0,1,2,···.Here p(x)=1
and the interval is [0,π/2].
2
Let f (x) = x (1 − x) for 0 ≤ x ≤ π/2. We will expand f (x) in a series ∞ c n ϕ n (x) of f
n=0
of the eigenfunctions of this problem, using equation (15.4) for the coefficients. First compute
π/2
f (x)ϕ 0 (x)dx
0
c 0 =
π/2 2
ϕ dx
0 0
π/2 2
0 x (1 − x)dx
=
π/2
dx
0
1 4 1 3
− π + π 1 1
3
2
= 64 24 =− π + π .
π 32 12
2
For n = 1,2,···, the denominator of c n is
π/2
2 π
cos (2nx)dx = .
0 4
The numerator is
π/2 n 2 2 2
2 −6 + (−1) [6 + 4πn − 3π n ]
x (1 − x)cos(2nx)dx = .
16n 4
0
Therefore
n 2 2 2
−6 + (−1) [6 + πn − 3π n ]
c n = .
4πn 4
The eigenfunction expansion is

1 1
2
2
x (1 − x) =π − π
12 32
∞ n 2 2 2
−6 + (−1) [6 + πn − 3π n ]
+ cos(2nx).
4πn 4
n=1
2
From the convergence theorem, this eigenfunction expansion converges to x (1 − x) for 0 < x <
π/2.
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October 14, 2010 15:20 THM/NEIL Page-512 27410_15_ch15_p505-562