196
4. Series-Expansion Methods
and the fonnula that results after substituting (4.42) into (4.41) is valid for Im I ::s
n. Note that when m is odd" the associated Legendre functions are not polynomiais in J-L.
The leading factor in (4.41) nonnalizes Pm,n so that
1
1 e.;(J-L)Pm,s (J-L) dJ-L = s.:
-I
(4.43)
where 8 n ,s = I if n = sand is zero otherwise. As a consequence, the orthogonality relation for the spherical harmonics becomes
_I_li l
1f
2rr
Ym,n()'" J-L)Y r:s rx, J-L) d): dJ-L = 8m,r8n,s,
-I -1f
(4.44)
where Y r:s is the complex conjugate of Yr,s. The associated Legendre functions
have the property that P-m,n (J-L) = (_1)m Pm,n(J-L), which implies that Lm,n =
(_1)my,;; n' and thus the expansion coefficients for any approximation to a realvalued
satisfy
a_m,n = (_1)m
(4.45)
Two recurrence relations satisfied by the associated Legendre functions that
will be used in the subsequent analysis are
and
J-LPm ,n = Em.n+IPm.n+1 + Em ,nPm,n-1
2 dPm ,n
(l - J-L
= -nEm,n+IPm,n+1 + (n + I)E
(4.46)
(4.47)
where
n 2 -m 2) 1/2
Em,n = ( 4n2 - I
The spherical harmonics are eigenfunctions of the Laplacian operator on the
sphere such that
V2y
_ -n(n + I) Y
(4.48)
m,n -
a 2
m,n,
where a is the radius of the sphere and the horizontal Laplacian operator in spherical coordinates is
7Specialized tenninology is sometimes used to differentiate between the indices of the associated
Legendre function Pm.n - The m index indicates the "order,' whereas the n index indicates the "degree ,"