The_Cambridge_Handbook_of_Physics_Formulas

48 Mathematics
Associated Legendre functions a
Associated d
Legendre
[(1−x 2 ) dP m
dx
equation
l
(x)
dx
]
]
+
[l(l +1)− m2
1−x 2 Pl m (x)=0
(2.428)
P m
l
associated
Legendre
functions
From
Legendre
polynomials
P m
l (x)=(1−x 2 ) m/2
Pl
−m (x)=(−1)
dm
dx m P l(x), 0 ≤ m ≤ l (2.429)
m (l −m)!
(l +m)! P l m (x) (2.430)
P l
Legendre
polynomials
Recurrence
relations
Orthogonality
P m m+1(x)=x(2m+1)P m m (x) (2.431)
Pm m (x)=(−1) m (2m−1)!!(1−x 2 ) m/2 (2.432)
(l −m+1)Pl+1(x)=(2l m +1)xPl m (x)−(l +m)Pl−1(x)
m
(2.433)
∫ 1
P m
−1
!! 5!! = 5·3·1 etc.
l (x)Pl m (l +m)! 2
′ (x)dx =
(l −m)! 2l +1 δ ll ′ (2.434) δ ll ′ Kronecker
delta
P 0 0 (x)=1 P 0 1 (x)=x P 1 1 (x)=−(1−x 2 ) 1/2
P 0 2 (x)=(3x 2 −1)/2 P 1 2 (x)=−3x(1−x 2 ) 1/2 P 2 2 (x)=3(1−x 2 )
a Of the first kind. P m
l (x) can be defined with a (−1) m factor in Equation (2.429) as well as Equation (2.430).
1
0.5
0
−0.5
−1
Legendre polynomials
P 2
P 0
P 3 P 4
P5
P 1
3
2
1
0
−1
associated Legendre functions
P 0 2
P 1 2
P 0 0
P 0 1
P 2 2
P 1 1
−1
−0.5
0
x
0.5
1
−1
0
x
1