Legendre Differential Equation

The second-order Ordinary Differential Equation

 (1)

which can be rewritten
 (2)

The above form is a special case of the associated Legendre differential equation with . The Legendre differential equation has Regular Singular Points at , 1, and . It can be solved using a series expansion,
 (3) (4) (5)

Plugging in,

 (6)

 (7)
 (8)
 (9)

 (10)

so each term must vanish and
 (11)

 (12)

Therefore,
 (13) (14) (15)

so the Even solution is

 (16)

Similarly, the Odd solution is

 (17)

If is an Even Integer, the series reduces to a Polynomial of degree with only Even Powers of and the series diverges. If is an Odd Integer, the series reduces to a Polynomial of degree with only Odd Powers of and the series diverges. The general solution for an Integer is given by the Legendre Polynomials

 (18)

where is chosen so that . If the variable is replaced by , then the Legendre differential equation becomes
 (19)

as is derived for the associated Legendre differential equation with .

The associated Legendre differential equation is

 (20)

 (21)

The solutions to this equation are called the associated Legendre polynomials. Writing , first establish the identities
 (22)

 (23)

 (24)

and
 (25)

Therefore,
 (26)

Plugging (22) into (26) and the result back into (21) gives

 (27)

 (28)

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 332, 1972.