Page 491 - Electromagnetics

Since the eigenfunctions are also complete, we can expand any piecewise continuous
function f in a Fourier–Bessel series

∞
x
f (x) = c m J ν p νm , 0 ≤ x ≤ L,ν > −1.
L
m=1
By (A.85) and (E.22) we have
2 L x
c m = f (x)J ν p νm xdx.
L J 2 (p νm ) L
2
ν+1 0
The associated Legendre equation. Legendre’s equation occurs when problems are
solved in spherical coordinates. It is often written in one of two forms. Letting θ be the
polar angle of spherical coordinates (0 ≤ θ ≤ π), the equation is
d dψ(θ) m 2
sin θ + λ sin θ − ψ(θ) = 0.
dθ dθ sin θ
2
This is Sturm–Liouville with p(θ) = sin θ, σ(θ) = sin θ, and q(θ) =−m / sin θ. The
boundary conditions

|ψ(0)| < ∞ and |ψ(π)| < ∞
define a singular problem: the conditions are not homogeneous, p(θ) = 0 at both end-
points, and q(θ) < 0. Despite this, the Legendre problem does share properties of a
regular Sturm–Liouville problem — including eigenfunction orthogonality and complete-
ness.
Using x = cos θ, we can put Legendre’s equation into its other common form
d 2 dψ(x) m 2
[1 − x ] + λ − ψ(x) = 0, (A.92)
dx dx 1 − x 2
where −1 ≤ x ≤ 1. It is found that ψ is bounded at x =±1 only if
λ = n(n + 1)

where n ≥ m is an integer. These λ are the eigenvalues of the Sturm–Liouville problem,
and the corresponding ψ n (x) are the eigenfunctions.
As a second-order partial differential equation, (A.92) has two solutions known as
associated Legendre functions. The solution bounded at both x =±1 is the associated
m
Legendre function of the first kind, denoted P (x). The second solution, unbounded at
n
m
x =±1, is the associated Legendre function of the second kind Q (x). Appendix E.2
n
tabulates some properties of these functions.
m
m
For fixed m, each λ mn is associated with a single eigenfunction P (x). Since P (x) is
n n
bounded at x =±1, and since p(±1) = 0, the eigenfunctions obey Lagrange’s identity
(A.79), hence are orthogonal on [−1, 1] with respect to the weight function σ(x) = 1.
Evaluation of the orthogonality integral leads to
1
m m 2 (n + m)!
P (x)P (x) dx = δ ln (A.93)
l
n
−1 2n + 1 (n − m)!
or equivalently
π
m m 2 (n + m)!
P (cos θ)P (cos θ) sin θ dθ = δ ln .
l n
0 2n + 1 (n − m)!
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