Page 510 - Electromagnetics

2
We introduce a new constant k to separate r from θ:
θ
1 d 2 dR(r) 2 2 2
r + k r = k , (A.139)
θ
R(r) dr dr
1 d d%(θ) µ 2 2
− sin θ + 2 = k . (A.140)
θ
sin θ%(θ) dθ dθ sin θ
Equation (A.140),
1 d d%(θ) 2 µ 2
sin θ + k − %(θ) = 0,
θ
2
sin θ dθ dθ sin θ
can be put into a standard form by letting

η = cos θ (A.141)
2
and k = ν(ν + 1) where ν is a parameter:
θ
2
d %(η) d%(η) µ 2
2
(1 − η ) − 2η + ν(ν + 1) − %(η) = 0, −1 ≤ η ≤ 1.
dη 2 dη 1 − η 2
This is the associated Legendre equation. It has two independent solutions called as-
µ
µ
sociated Legendre functions of the first and second kinds, denoted P (η) and Q (η),
ν
ν
respectively. In these functions, all three quantities µ,ν,η may be arbitrary complex
constants as long as ν + µ = −1, −2,.... But (A.141) shows that η is real in our discus-
sion; µ will generally be real also, and will be an integer whenever #(φ) is single-valued.
µ
The choice of ν is somewhat more complicated. The function P (η) diverges at η =±1
ν
µ
unless ν is an integer, while Q (η) diverges at η =±1 regardless of whether ν is an inte-
ν
µ
ger. In § A.4 we required that P (η) be bounded on [−1, 1] to have a Sturm–Liouville
ν
µ
problem with suitable orthogonality properties. By (A.141) we must exclude Q (η) for
ν
µ
problems containing the z-axis, and restrict ν to be an integer n in P (η) for such prob-
ν
lems. In case the z-axis is excluded, we choose ν = n whenever possible, because the finite
m
µ
m
µ
sums P (η) and Q (η) are much easier to manipulate than P (η) and Q (η). In many
n n ν ν
0
problems we must count on completeness of the Legendre polynomials P n (η) = P (η) or
n
spherical harmonics Y mn (θ, φ) in order to satisfy the boundary conditions. In this book
we shall consider only those boundary value problems that can be solved using integer
values of ν and µ, hence choose
m
m
%(θ) = A θ P (cos θ) + B θ Q (cos θ). (A.142)
n
n
Single-valuedness in #(φ) is a consequence of having µ = m, and φ = constant boundary
surfaces are thereby disallowed.
The associated Legendre functions have many important properties. For instance,

0, m > n,
m
2
2 m/2
P (η) = m (1 − η ) d n+m (η − 1) n (A.143)
n
(−1) n n+m , m ≤ n.
2 n! dη
The case m = 0 receives particular attention because it corresponds to azimuthal invari-
0
ance (φ-independence). We define P (η) = P n (η) where P n (η) is the Legendre polynomial
n
© 2001 by CRC Press LLC