Eduardo Kausel-Fundamental solutions in elastodynamics_ a compendium-Cambridge University Press (2006)

190 Basic properties of mathematical functions
11.3.2 Rodrigues’s formula
P m (x) = (−1)m d m (1 − x 2 ) m
,
2 m m! dx m P 0 (x) = 1, P 1 (x) = x, P 2 (x) = 1 2 (3x2 − 1),...
(11.31)
Observe that P m (1) = 1 and P m (−1) = (−1) m .
11.3.3 Trigonometric expansion (x = cos φ)
[
(2m − 1)!!
P m (φ) = 2 cos mφ + 1 m
cos(m − 2)φ
(2m)!!
1 (2m − 1)
+ 1 × 3 m(m − 1)
cos(m − 4)φ +···]
1 × 2 (2m − 1) (2m − 3)
(11.32)
Divide the last term by 2 if miseven! Here, (2m − 1)!! = 1 × 3 × 5 ×···×(2m − 1)
and (2m)!! = 2 × 4 × 6 ×···×2m. The first six Legendre polynomials in trigonometric
form are
P 0 = 1, P 1 = cos φ, P 2 = 1 4 (3 cos 2φ + 1) , P 3 = 1 (5 cos 3φ + 3 cos φ) (11.33)
8
P 4 = 1 64 (35 cos 4φ + 20 cos 2φ + 9) , P 5 = 1 (63 cos 5φ + 35 cos 3φ + 30 cos φ)
128
(11.34)
11.3.4 Recurrence relations
(m + 1)P m+1 = (2m + 1)x P m − m P m−1 (simplest way to find P m ) (11.35)
(1 − x 2 ) dP m
dx = m(P m(m + 1)
m−1 − x P m) =
2m + 1 (P m−1 − P m+1) (11.36)
sin φ dP m
dφ = m(cos φ P m(m + 1)
m − P m−1) =
2m + 1 (P m+1 − P m−1) (11.37)
11.3.5 Orthogonality condition
∫ +1
−1
P m (x) P n (x) dx =
2
2m + 1 δ mn (11.38)
In its trigonometric form, with x = cos φ, the orthogonality condition is
∫ π
0
P m (cos φ) P n (cos φ) sin φ dφ =
2
2m + 1 δ mn (11.39)