The Chemistry Maths Book, Second Edition

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

The Legendre polynomials satisfy the recurrence relation

(l 1 + 1 1)P

l+ 1

(x) 1 − 1 (2l 1 + 1 1)xP

l

(x) 1 + 1 lP

l− 1

(x) 1 = 10 (13.21)

so that givenP

0

(x) 1 = 11 andP

1

(x) 1 = 1 x, all higher polynomials can be derived.

EXAMPLE 13.6Use the recurrence relation (13.21) to findP

2

(x)andP

3

(x).

Forl 1 = 11 the relation (13.21) is 2 P

2

1 − 13 xP

1

1 + 1 P

0

1 = 10. Therefore,

Forl 1 = 12 the relation (13.21) is 3 P

3

1 − 15 xP

2

1 + 12 P

1

1 = 10. Therefore,

0 Exercise 17

The associated Legendre functions

The associated Legendre equation is (see Table 3.1)

(13.22)

As with the Legendre equation (m 1 = 10 ), the variable xis identified withcos 1 θin

physical applications. In this case, the solutions obtained by the power-series method

converge in the interval− 11 ≤ 1 x 1 ≤ 11 when both land mare integers, with| 1 m 1 | 1 ≤ 1 l:

l 1 = 1 0, 1, 2, 3, = m 1 = 1 0, ±1, ±2, =, ±l (13.23)

The corresponding particular solutions are called the associated Legendre functions

P

l

|m|

(x). They are related to the Legendre polynomials by the differential formula

(13.24)

withP

l

0

1 = 1 P

l

.

EXAMPLE 13.7Use the formula (13.24) to derive the associated Legendre functions

P

m

3

for m 1 = 1 1, 121 and 1 3, and express these as functions of θwhen x 1 = 1 cos 1 θ and

(1 1 − 1 x

2

)

122

1 = 1 sin 1 θ.

Px x

d

dx

Px

l

mm

m

m

l

||

||

||

() ( )=− ()

||

1

22

() ()

()

12 1

1

0

2

2

2

− ′′− ′++−

−

















xy xy ll =

m

x

y

PxPP xx x

321

2

1

3

52

1

3

5

1

2

312

1

2

=−=×−−













() ()(5= 53

3

xx− )

PxPP x

210

2

1

2

3

1

2

=−=−()() 31