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

11.3 Legendre polynomials 189
1.0
P2
P
0.5
3
P P 5 4
0
-0.5
P 2
-1.0
0 0.2 0.4 0.6 0.8 1.0
1
π φ
Figure 11.3: Legendre polynomials.
The functions j m satisfy j m (0) = δ m0 , that is, they vanish at the origin except for j 0 . Also,
the y m functions are singular at z = 0. More generally,
( ) 1
j m (z) = (−z) m d m ( (
sin z 1
= (−z) m d 1 d ···( )))
1 d sin z
, m = 0, 1, 2,...
z dz z
z dz z dz z dz z
(11.25)
( ) 1
y m (z) =−(−z) m d m
cos z
z dz z
(11.26)
11.2.3 Recurrence relations
s m (z) = any of the spherical Bessel functions
2m + 1
s m (z) = s m−1 (z) + s m+1 (z)
z
(11.27)
(2m + 1) ds m(z)
= ms m−1 (z) − (m + 1) s m+1 (z)
dz
(11.28)
11.3 Legendre polynomials
11.3.1 Differential equation
(1 − x 2 ) d2 y dy
− 2x + m(m + 1)y = 0, m = 0, 1, 2,..., |x| ≤ 1 (11.29)
dx2 dx
For non-negative integer m, the solution to this equation is y = c 1 P m + c 2 Q m , with c 1 , c 2
being arbitrary constants, and P m (x) the Legendre polynomial of order m. The second
solution Q m (x) to the Legendre differential equation is used much less often. In most
applications, x = cos φ. The trigonometric form of the differential equation is
d 2 P m
dφ 2
+ cot φ dP m
dφ + m(m + 1) P m = 0 (11.30)