Figure Properties - SERC

legendre
2legendre
Purpose Associated Legendre functions
Syntax P = legendre(n,X)
S = legendre(n,X,'sch')
N = legendre(n,X,'norm')
Definitions Associated Legendre Functions. The Legendre functions are defined by
2-1298
m
P ( x)
( – 1)
m( 1 – x2) m 2
n
where
/
xm m
d
=
Pn( x)
d
Pn( x)
is the Legendre polynomial of degree n .
Pn( x)
1
2 n ----------n!
dn
dx n
--------- x 2 ( – 1)
n
=
Schmidt Seminormalized Associated Legendre Functions. The Schmidt seminormalized
associated Legendre functions are related to the nonnormalized associated
m
Legendre functions Pn ( x)
by
Pn( x)
for m = 0
m m
Sn ( x)
( – 1)
2( n – m)!
m
= ------------------------ P for .
( n + m)!
n ( x)
m > 0
Fully Normalized Associated Legendre Functions. The fully normalized associated
Legendre functions are normalized such that
1
m 2
∫ ( Nn ( x)
) dx = 1
– 1
m
and are related to the unnormalized associated Legendre functions Pn ( x)
by