Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
We ®nd successively
a 2 ˆ
a 4 ˆ
ˆ
… ‡ 1†
… 1†… ‡ 2†
a
2! 0 ; a 3 ˆ a
3! 1 ;
… 2†… ‡ 3†
… 3†… ‡ 4†
a
4 3 2 ; a 5 ˆ a
3! 3 ;
… 2†… ‡ 1†… ‡ 3†
a
4!
0 ; ˆ
… 3†… 1†… ‡ 2†… ‡ 4†
a
5!
1 ;
etc. By inserting these values for the coecients into Eq. (7.2) we obtain
y…x† ˆa 0 y 1 …x†‡a 1 y 2 …x†;
…7:6†
where
y 1 …x† ˆ1
… ‡ 1†
x 2 ‡
2!
… 2†… ‡ 1†… ‡ 3†
x 4 ‡
4!
…7:7a†
and
y 2 …x† ˆx ˆ
… 1†… ‡ 2†
x 3 ‡
3!
… 2†… 1†… ‡ 2†… ‡ 4†
x 5 ‡: …7:7b†
5!
These series converge for jxj < 1. Since Eq. (7.7a) contains even powers of x, and
Eq. (7.7b) contains odd powers of x, the ratio y 1 =y 2 is not a constant, and y 1 and
y 2 are linearly independent solutions. Hence Eq. (7.6) is a general solution of Eq.
(7.1) on the interval 1 < x < 1.
In many applications the parameter in Legendre's equation is a positive
integer n. Then the right hand side of Eq. (7.5) is zero when s ˆ n and, therefore,
a n‡2 ˆ 0 and a n‡4 ˆ 0; ...: Hence, if n is even, y 1 …x† reduces to a polynomial of
degree n. Ifn is odd, the same is true with respect to y 2 …x†. These polynomials,
multiplied by some constants, are called Legendre polynomials. Since they are of
great practical importance, we will consider them in some detail. For this purpose
we rewrite Eq. (7.5) in the form
…s ‡ 2†…s ‡ 1†
a s ˆ
…ns†…n ‡ s ‡ 1† a s‡2
…7:8†
and then express all the non-vanishing coecients in terms of the coecient a n of
the highest power of x of the polynomial. The coecient a n is then arbitrary. It is
customary to choose a n ˆ 1 when n ˆ 0 and
a n ˆ …2n†! 1 3 5 …2n 1†
ˆ ; n ˆ 1; 2; ...; …7:9†
2 n 2
…n!† n!
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