3.3 Legendre Functions of Integer Type
41
from which we can derive directly the Jacobi matrix, see Chap. 15,
J P =
⎛
⎜
⎜
⎜
⎝
0
√
1/3
0
0 0· · ·
√
1/3
02
√
1/15
0 0 · · ·
0 2
√
1/15
0 3
√
1/35 0 · · ·
. . .
. . .
. . .
. . .
. . .
. . .
⎞
⎟
⎟
⎟
⎠
(3.8)
to compute the zeros of the Legendre polynomial of first kind. The nodes of the
Legendre polynomials are given by the eigenvalues of the corresponding Jacobi
matrix. Computing the nodes via eigenvalues is significantly more accurate than
computing the roots of the corresponding polynomials. The zeros “x0” of the
Legendre polynomial are returned by the tiny program x0 = JacobiPl(l) with
“l” the degree.
3.3
Legendre Functions of Integer Type
Associate Legendre functions of integer degree and integer order as well as spherical
harmonics play an important role in many areas in physics. For example, solutions
of the Laplace equation in spherical coordinates (23.5, 23.6b) can be written as
< r, θ, φ|Ψ >=
∞
l=0
l
m=−l
a lm r
l
+ b lm r
−(l+1)
Y
m
l (θ, φ),
(3.9a)
and in case of cylindrical symmetry, it simplifies to
< r, θ|Ψ >=
∞
l=0
a l r
l
+ b l r
−(l+1)
P l (cos θ) ,
(3.9b)
with a and b complex constants. Legendre polynomials P l have been discussed in
the last chapter. The θ -part of the spherical harmonics is given by the associate
Legendre functions. In the next two subsections, we will first discuss the associate
Legendre functions and as an application the spherical harmonics and in the next
section associate Legendre functions of first and second kinds with complex indices.
The conventions used are
• x ∈ R, −1 ≤ x ≤ 1
• cos(θ ) − π ≤ θ ≤ π
• z ∈ C
• indices n, l, m ∈ Z
• indices ν, μ ∈ C.