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or associated function):
r
(2 n − 2 k )!
P nm ( t )=2 −n (1
t 2 ) m/ 2
1) k
2 k )! t n−m− 2 k ,
(1-67)
(
k !( n
k )! ( n
m
k =0
where r is the greatest integer
( n
m ) / 2; i.e., r is ( n
m ) / 2or( n
m
1) / 2,
whichever is an integer. This formula is convenient for programming.
As this useful formula is seldom found in the literature, we show the
derivation, which is quite straightforward. The necessary information on fac-
torials may be obtained from any collection of mathematical formulas. The
binomial theorem gives
1) k n
k
t 2 n− 2 k =
n
n
n !
k !( n
( t 2
1) n =
1) k
k )! t 2 n− 2 k .
(
(
(1-68)
k =0
k =0
Thus, (1-57) becomes
n
d n + m
P nm ( t )= 1
1
k !( n
t 2 ) m/ 2
1) k
dt n + m ( t 2 n− 2 k ) ,
2 n (1
(
(1-69)
k )!
k =0
the quantity n ! having been cancelled out. The r th derivative of the power
t s is
d r
s !
dt r ( t s )= s ( s
r +1) t s−r =
r )! t s−r .
···
( s
1)
(1-70)
( s
Setting r = n + m and s =2 n − 2 k ,wehave
d n + m
(2 n − 2 k )!
( n
dt n + m ( t 2 n− 2 k )=
2 k )! t n−m− 2 k .
(1-71)
m
Inserting this into the above expression for P nm ( t ) and noting that the lowest
possible power of t is either t or t 0 = 1, we obtain (1-67).
The surface spherical harmonics are Legendre's functions multiplied by
cos or sin :
degree 0
P 0 (cos ϑ );
P 1 (cos ϑ ) ,
P 11 (cos ϑ )cos λ, P 11 (cos ϑ )sin λ ;
degree 1
(1-72)
degree 2
P 2 (cos ϑ ) ,
P 21 (cos ϑ )cos λ, P 21 (cos ϑ )sin λ,
P 22 (cos ϑ )cos2 λ, P 22 (cos ϑ )sin2 λ ;
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