Page 531 - Advanced engineering mathematics

15.1 Eigenfunction Expansions
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One proceeds in similar fashion for periodic and singular boundary conditions. This proves
conclusion (3) of the theorem.
For conclusion (4), let λ be any eigenvalue, with corresponding eigenfunction ϕ(x).Itis
routine to check by taking complex conjugates that λ is an eigenvalue with eigenfunction ϕ(x).
If λ = λ, then by conclusion (3),
b
(λ − λ) p(x)ϕ(x)ϕ(x)dx = 0.
a
Then
b
2
(λ − λ) p(x)|ϕ(x)| dx = 0.
a
But p(x)> 0on (a,b), and an eigenfunction cannot be identically zero, so this integral must be
positive. Therefore λ = λ, implying that λ must be real, proving conclusion (4).
The integral relationship (15.2) between eigenfunctions associated with distinct eigenvalues
is called orthogonality of the eigenfunctions with respect to the weight function p. This terminol-
b
ogy derives from the fact that p(x) f (x)g(x)dx behaves like a dot product for vectors, and
a
two vectors are called orthogonal when their dot product is zero. In Example 15.1, p(x) = 1 and
the orthogonality relationship is the familiar
L
sin(nπx/L)sin(mπx/L)dx = 0
0
for n = m.
Using this notion of weighted orthogonality of eigenfunctions, we can solve for the coef-
ficients in a proposed series of eigenfunctions, arguing much as we did for the coefficients of
a Fourier series. Suppose the eigenvalues are λ k and corresponding eigenfunctions are ϕ k for
k = 1,2,···, and we want to write
∞
f (x) = c k ϕ k (x).
k=1
Multiply this equation by p(x)ϕ n (x) and integrate to get
b ∞ b
p(x) f (x)ϕ n (x)dx = p(x)ϕ k (x)ϕ n (x)dx.
a k=1 a
By equation (15.2) all of the integrals in the summation are zero except when k = n, yielding
b b
2
p(x)ϕ (x)dx,
p(x) f (x)ϕ n (x)dx = c n n
a a
or
b
a p(x) f (x)ϕ n (x)dx
c n = . (15.4)
b
2
p(x)ϕ (x)dx
a n
By analogy with Fourier series, we call the numbers defined by equation (15.4) the general-
ized Fourier coefficients of f with respect to the Sturm-Liouville problem. With this choice of
coefficients, we call
∞
c k ϕ k (x)
k=1
the eigenfunction expansion of f with respect to the eigenfunctions of the Sturm-Liouville
problem.
As with Fourier (trigonometric) series, the question now is the relationship between the
function and its eigenfunction expansion.
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