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3 Legendre Polynomials and Legendre Functions
3.3.1 Fundamental Equations and Computation
The associate Legendre functions are the solutions of the Legendre equation
(1 − z
2 )
d 2
dz 2 − 2z
d
dz
+
ν(ν + 1) −
μ 2
1 − z 2
y(z) = 0,
(3.10)
where the indices “ν” and “μ” are referred to as the degree and order. For μ = 0
and ν = l integer, these functions are identical to the Legendre polynomials. For
−1 ≤ x ≤ +1, and ν = l, μ = m integers with 0 ≤ |m| ≤ l, these functions are
nonsingular and can be derived from the Legendre polynomials via
P
m
l (x) = (−1)
m (1 − x
2 )
m/2 d m
dx m P l (x) , and
(3.11a)
P
m
l (z) = (z
2
− 1)
m/2 d m
dz m P l (z) .
(3.11b)
Identical equations hold for the Legendre functions Q
m
l of second kind (P ↔ Q).
Q
μ
ν will be discussed in the next chapter.
The following relations between positive and negative orders are computationally
useful:
P
−m
l
(x) = (−1)
m (l − m)!
(l + m)!
P
m
l (x) , P
−m
ν (x) = (−1)
m Γ (ν − m + 1)
Γ (ν + m + 1)
P
m
ν (x)
(3.12a)
P
−μ
ν (z) =
Γ (ν − μ + 1)
Γ (ν + μ + 1)
P
μ
ν (z) −
2
π
exp(−iμπ) sin(μπ)Q
μ
ν (z)
(3.12b)
P
−μ
ν (x) =
Γ (ν − μ + 1)
Γ (ν + μ + 1)
cos(μπ)P
μ
ν (x) −
2
π
sin(μπ)Q
μ
ν (x)
(3.12c)
P
μ
−ν−1 (z) = P
μ
ν (z), P
μ
−ν−1 (x) = P
μ
ν (x)
(3.12d)
Q
−μ
ν (z) = exp(−2μπi)
Γ (ν − μ + 1)
Γ (ν + μ + 1)
Q
μ
ν (z)
(3.12e)
Q
−m
ν (x) = (−1)
m Γ (ν − m + 1)
Γ (ν + m + 1)
Q
m
ν (x).
(3.12f)