Introduction to Acoustics

70 Part A Propagation of Sound
Part A 3.11
–1
P4
P3
P2
P0
–1
Pl(�)
Fig. 3.25 Legendre polynomials for various orders
1
The latter holds for ℓ ≥ 1. With this, for example, given
that P0(ξ) = 1andthatP1(ξ) = ξ, one derives
(2)P2(ξ) = (3)ξξ − (1)(1) . (3.375)
The first few of these polynomials are
P0(ξ) = 1 , (3.376)
P1(ξ) = ξ, (3.377)
P2(ξ) = 1
2 (3ξ2 − 1) , (3.378)
P3(ξ) = 1
2 (5ξ3 − 3ξ) , (3.379)
P4(ξ) = 1
8 (35ξ4 − 30ξ 2 + 3) , (3.380)
P5(ξ) = 1
8 (63ξ5 − 70ξ 3 + 15ξ) , (3.381)
with the customary identification of ξ = cos θ.
An alternate statement for the series expression
(3.371), given Kℓ = 1, is the Rodrigues relation,
Pℓ(ξ) = 1
2ℓ d
ℓ!
ℓ
dξℓ (ξ2 − 1) ℓ . (3.382)
This can be verified by using the binomial expansion
(ξ 2 − 1) ℓ = (−1) ℓ
P1
ℓ�
(−1) n ℓ!
n!(ℓ − n)! ξ2n , (3.383)
n=1
1
�
so that
dℓ dξℓ (ξ2 − 1) ℓ =(−1) ℓ
ℓ�
(−1)
n=ℓ−M
n ℓ!
n!(ℓ − n)!
× (2n)!
(2n − ℓ)! ξ2n−ℓ , (3.384)
or, with the change of summation index to m = ℓ − n,
dℓ dξℓ (ξ2 − 1) ℓ M(ℓ) �
= (−1) m
m=0
× (2ℓ − 2m)!
ℓ!
m!(ℓ − m)!
(ℓ − 2m)! ξℓ−2m = ℓ!2 ℓ Pℓ(ξ) .
(3.385)
Another derivable property of these functions is that
they are orthogonal in the sense that
�π
0
Pℓ(cos θ)Pℓ ′(cos θ)sinθ dθ = 0 if ℓ �= ℓ′ .
(3.386)
This is demonstated by taking the differential equations
(3.361) satisfied by Pℓ and Pℓ ′, multiplying the first
by Pℓ ′ sin θ, multiplying the second by Pℓ(θ)sinθ, then
subtracting the second from the first, with a subsequent
integration over θ from 0 to π. Given that λℓ �= λℓ ′ and
that the two polynomials are finite at the integration
limits, the conclusion is as stated above.
If the two indices are equal, the chosen normalization,
whereby Pℓ(1) = 1, leads to
�π
0
[Pℓ(cos θ)] 2 sin θ dθ = 2
. (3.387)
2ℓ + 1
The general derivation of this makes use of the Rodrigues
relation (3.382) and of multiple integrations by
parts. To carry through the derivation, one must first verify,
for arbitrary nonnegative integers s and t, and with
s < t,that
d
dξ s (ξ2 − 1) t = 0 atξ =±1 , (3.388)
which is accomplished by use of the chain rule of differential
calculus. With the use of this relation and of the