The Chemistry Maths Book, Second Edition

376 Chapter 13Second-order differential equations. Some special functions

Similarly for the odd values of m,

A power-series solution of the equation is therefore

y(x) 1 = 1 a

0

y

1

(x) 1 + 1 a

1

y

2

(x) (13.16)

where a

0

and a

1

are arbitrary constants and

(13.17)

(13.18)

The seriesy

1

contains only even powers of x, the seriesy

2

only odd powers.

Convergence

We first consider the case of non-integer values of l. Both series have radius of

convergenceR 1 = 11 for arbitrary values of l. Thus, the ratio of consecutive terms is,

by equation (13.15),

so that, by d’Alembert’s ratio test (7.18) for power series, both series converge if| 1 x 1 | 1 < 11

and diverge if| 1 x 1 | 1 > 11 (unless lis an integer, see below). Both series also diverge when

| 1 x 1 | 1 = 11. The function (13.16), with two arbitrary constants, is therefore the general

solution of the Legendre equation in the interval− 11 < 1 x 1 < 11. It is also possible to find

a solution involving inverse powers of xthat is valid for| 1 x 1 | 1 > 11.

The Legendre polynomials

The function (13.17) reduces to a polynomial of degree lwhen lis an even integer or

zero. For example, whenl 1 = 12 the series terminates after the second term to give

y

1

(x) 1 = 111 − 13 x

2

. The choicea

1

1 = 10 in (13.16) therefore gives a particular solution for

even values of lthat is finite for all values of x. Similarly, the function (13.18) reduces

to a polynomial of degree lwhen lis an odd integer, and the choicea

0

1 = 10 in (13.16)

gives a particular solution that is valid for all values of x. The same considerations

apply for negative integer values of l, but the resulting set of polynomials is identical

to that obtained for positive values. These particular solutions are called Legendre

polynomialsP

l

(x), and they are the solutions of the Legendre equation that are of

a

a

mm ll

mm

m

m

m

+

=

+− +

++

→→

2

11

12

1

()()

()( )

as ∞

yx x

ll

x

llll

2

3

12

3

3124

()

()() ()()()()

=−

−+

!

• 
  

−−+ +

55

5

!

x −

yx

ll

x

lll l

x

1

24

1

1

2

213

4

()

() ()()()

=−

• 
  

!

• 
  

−++

!

−

a

ll

aa

llll

315

12

3

3124

=−

−+

!

,=+

()() ()()()()−−+ +

55

1

!

a, ...