Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
We could solve Eq. (7.25) by series; but it is more useful to know how the
solutions are related to Legendre polynomials, so we shall proceed in the following
way. We write
y ˆ…1 x 2 † m=2 u…x†
and substitute into Eq. (7.25) whence we get, after a little simpli®cation,
…1 x 2 †u 00 2…m ‡ 1†xu 0 ‡‰n…n ‡ 1†m…m ‡ 1†Šu ˆ 0: …7:26†
For m ˆ 0, this is a Legendre equation with solution P n …x†. Now we di€erentiate
Eq. (7.26) and get
…1 x 2 †…u 0 † 00 2‰…m ‡ 1†‡1Šx…u 0 † 0 ‡‰n…n ‡ 1†…m ‡ 1†…m ‡ 2†Šu 0 ˆ 0:
…7:27†
Note that Eq. (7.27) is just Eq. (7.26) with u 0 in place of u, and (m ‡ 1) in place of
m. Thus, if P n …x† is a solution of Eq. (7.26) with m ˆ 0, Pn…x† 0 is a solution of Eq.
(7.26) with m ˆ 1, Pn 00 …x† is a solution with m ˆ 2, and in general for integral
m; 0 m n; …d m =dx m †P n …x† is a solution of Eq. (7.26). Then
y ˆ…1 x 2 † m=2 d m
dx m P n…x†
…7:28†
is a solution of the associated Legendre equation (7.25). The functions in Eq.
(7.28) are called associated Legendre functions and are denoted by
P m n …x† ˆ…1 x2 † m=2 d m
dx m P n…x†:
…7:29†
Some authors include a factor (1† m in the de®nition of P m n …x†:
A negative value of m in Eq. (7.25) does not change m 2 , so a solution of Eq.
(7.25) for positive m is also a solution for the corresponding negative m. Thus
many references de®ne P m n …x† for n m n as equal to Pn
jmj …x†.
When we write x ˆ cos , Eq. (7.25) becomes
and Eq. (7.29) becomes
In particular
( )
1 d dy
sin
‡ n…n ‡ 1† m2
sin d d
sin 2
d m
P m n …cos † ˆsin m
d…cos † m fP n …cos † g:
D 1 means
308
Z x
1
P n …x†dx:
y ˆ 0
…7:30†