Page 529 - Advanced engineering mathematics

15.1 Eigenfunction Expansions
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we ask whether it might be possible to expand an “arbitrary” function f on [a,b] in a series of
these eigenfunctions,
∞
f (x) = c k ϕ k (x).
k=1
The key lies in the choice of the coefficients c k , which in turn hinges on an orthogonality property
of the eigenfunctions. Recall that, in seeking the coefficients a n and b n in the Fourier expansion
∞
1
f (x) = a 0 + (a k cos(kπx/L) + b k sin(kπx/L))
2
k=1
we multiplied the series by a particular eigenfunction cos(nπx/L) or sin(nπx/L) and integrated.
Because of equations (13.3), (13.4) and (13.5), the integral of any two distinct eigenfunctions
vanished, leaving simple integral expressions for a n or b n .
We will show that similar properties hold for the eigenfunctions of any Sturm-Liouville
problem, suggesting a similar strategy for finding the coefficients in a series of eigenfunctions
∞
k=1 c k ϕ k (x).
We will also show the eigenvalues must be real numbers. Thus the absence of complex
eigenvalues in Example 15.1 and 15.2 is characteristic of Sturm-Liouville problems in general.
THEOREM 15.1
Suppose we have a regular, periodic or singular Sturm-Liouville problem on an interval [a,b].
Then
1. There is an infinite sequence of eigenvalues λ j which can be ordered so that
λ 1 <λ 2 < ··· .
Further, with this ordering as an increasing sequence,
lim λ n =∞.
n→∞
2. If ϕ is an eigenfunction, then so is cϕ for any nonzero real number c.
3. Let λ n and λ m be distinct eigenvalues of the problem, with corresponding eigenfunctions
ϕ n and ϕ m . Then
b
p(x)ϕ n (x)ϕ m (x)dx = 0, (15.2)
a
where p is the coefficient of λ in the Sturm-Liouville differential equation (ry ) +
(q + λp)y = 0.
4. Every eigenvalue of the Sturm-Liouville problem is real.
Proof Conclusion (1) requires a lengthy analysis we will not enter into here. Note that con-
clusion (1) implies that the eigenvalues cannot cluster around a finite number, as, for example,
the numbers 1 − 1/n fall within arbitrarily small intervals about 1 as n is chosen larger. These
numbers cannot be the eigenvalues of a Sturm-Liouville problem.
For conclusion (2), suppose ϕ(x) is an eigenfunction corresponding to eigenvalue λ. Then
(rϕ ) + (q + λp)ϕ = 0.
Multiplication of this equation by c shows that cϕ satisfies this differential equation as well.
Finally, by multiplying the appropriate boundary conditions in each case by c, it is verified that
cϕ is an eigenfunction corresponding to λ.(Note: cϕ is an eigenfunction corresponding to λ, not
to cλ.)
October 14, 2010 15:20 THM/NEIL Page-509 27410_15_ch15_p505-562