THE ASSOCIATED LEGENDRE FUNCTIONS

10. THE ASSOCIATED LEGENDRE FUNCTIONS

A differential equation closely related to the Legendre equation is

(10.1) (1 − x 2 )y ′′

− 2xy + l(l + 1) −

2 1−x y=0

with m 2 ≤l 2 . We could solve this equation by series; however, it is more useful to know how the solutions are related to Legendre polynomials, so we shall simply verify the known solution. First we substitute

y = (1 − x 2 ) m/2 u

into (10.1) and obtain (Problem 1) (10.3)

(1 − x 2 )u ′′ − 2(m + 1)xu ′ + [l(l + 1) − m(m + 1)]u = 0.

For m = 0, this is Legendre’s equation with solutions P l (x). Differentiate (10.3), obtaining (Problem 1)

(10.4) (1 − x 2 )(u ′ ) ′′ − 2[(m + 1) + 1]x(u ′ ) ′ + [l(l + 1) − (m + 1)(m + 2)]u ′ = 0. But this is just (10.3) with u ′ in place of u, and (m + 1) in place of m. In other

words, if P l (x) is a solution of (10.3) with m = 0, P ′ l (x) is a solution of (10.3) with m = 1, P l ′′ (x) is a solution with m = 2, and in general for integral m, 0 ≤ m ≤ l, (d m /dx m )P l (x) is a solution of (10.3). Then

is a solution of (10.1). The functions in (10.5) are called associated Legendre func- tions are are denoted by

(10.6) P m

Associated Legendre functions

dx m

[Some authors include a factor (−1) m in the definition of P m l (x).]

A negative value for m in (10.1) does not change m 2 , so a solution of (10.1) for positive m is also a solution for the corresponding negative m. Thus many references define P m

l (x) for −l ≤ m ≤ l as equal to P l (x). Alternatively, we may use Rodrigues’ formula (4.1) for P l (x) in (10.6) to get

|m|

1 d l+m

(10.7) P m l (x) =

l+m (x − 1) 2 . l! dx It can be shown that (10.7) is a solution of (10.1) for either positive or nega-

2 l (1 − x ) m/2

tive m; however P −m l (x) and P m l (x) are then proportional rather than equal (see Problem 8).

584 Series Solutions of Differential Equations Chapter 12

For each m, the functions P m l (x) are a set of orthogonal functions on (−1, 1) (Problem 3). The normalization constants can be evaluated; for the definition (10.7) we find (Problem 10)

2 2 (10.8) (l + m)! [P

l (x)] dx =

2l + 1 (l − m)!

The associated Legendre functions arise in many of the same problems in which Legendre polynomials appear (see the first paragraph of Section 2); in fact, the Legendre polynomials are just the special case of the functions P m l (x) when m = 0.

PROBLEMS, SECTION 10

1. Verify equations (10.3) and (10.4). 2. The equation for the associated Legendre functions (and for Legendre functions

when m = 0) usually arises in the form (see, for example, Chapter 13, Section 7)

Make the change of variable x = cos θ, and obtain (10.1). 3. Show that the functions P l m (x) for each m are a set of orthogonal functions on

(-1,1), that is, show that

Z 1 P l m (x)P n m (x) dx = 0, −1

Hint: Use the differential equations (10.1) and follow the method of Section 7. Substitute the P l (x) you found in Problems 4.3 or 5.3 into equation (10.6) to find P m l (x);

then let x = cos θ to evaluate: 1 1 4. 2 P

1 (cos θ)

5. P 4 (cos θ)

6. P 3 (cos θ)

7. Show that

− 1) l = (x − 1) (x + 1) l and find the derivatives by Leibniz’ rule.

Hint: Write (x 2 l

8. Write (10.7) with m replaced by −m; then use Problem 7 to show that

P l (x).

(l + m)!

Comment: This shows that (10.7) is a solution of (10.1) when m is negative. 9. Use Problem 7 to show that

m (l + m)! (1 − x 2 ) −m/2 d P l−m

(x) = (−1)

(x 2 − 1) .

dx l−m 10. Derive (10.8) as follows: Multiply together the two formulas for P m l (x) given in

(l − m)!

2 l l!

(10.7) and Problem 9. Then integrate by parts repeatedly lowering the l + m deriva- tive and raising the l − m derivative until both are l derivatives. Then use (8.1).

Section 11 Generalized Power Series or the Method of Frobenius 585