Mathematical Tools for Physics

4—Differential Equations 109

The numerator in Eq. ( 37 ) foran+2is[(n+s)(n+s+ 1)−C]. If this happen to equal zero for some value
ofn=N, thenaN+2= 0and so then all the rest ofaN+4...are zero too. The series is a polynomial. This
will happen only for special values ofC, such as the valueC= 6above. The values ofCthat have this special
property are
C=(+ 1), for `= 0, 1 , 2 , ... (40)

This may be easier to see in the explicit representation, Eq. ( 39 ). When a numerator equals zero, all the rest that
follow are zero too. WhenC=(+ 1)for even, the first series terminates in a polynomial. Similarly for odd the second series is a polynomial. These are the Legendre polynomials, denotedP`(x), and the conventional
normalization is to require that their value atx= 1is one.

P 0 (x) = 1 P 1 (x) =x P 2 (x) =^3 / 2 x^2 −^1 / 2

P 3 (x) =^5 / 2 x^3 −^3 / 2 x P 4 (x) =^35 / 8 x^4 −^30 / 8 x^2 +^3 / 8

(41)

The special case for which the series terminates in a polynomial is by far the most commonly used solution to
Legendre’s equation. You seldom encounter the general solutions of Eq. ( 39 ).
A few properties of theP`are

(a)

(b)

(c)

(d)

∫ 1

− 1

dxPn(x)Pm(x) =

2

2 n+ 1

δnm where δnm=

{

1 ifn=m

0 ifn 6 =m

(n+ 1)Pn+1(x) = (2n+ 1)xPn(x)−nPn− 1 (x)

Pn(x) =

(−1)n

2 nn!

dn

dxn

(1−x^2 )n

Pn(1) = 1 Pn(−1) = (−1)n

(42)