Main points of Legendre differential equation?

Answer 1

The Legendre differential equation is the second-order ordinary differential equation(1)

which can be rewritten(2)

The above form is a special case of the so-called "associated Legendre differential equation" corresponding to the case . The Legendre differential equation has regular singular points at , 1, and .

If the variable is replaced by , then the Legendre differential equation becomes(3)

derived below for the associated () case.

Since the Legendre differential equation is a second-order ordinary differential equation, it has two linearly independent solutions. A solution which is regular at finite points is called a Legendre function of the first kind, while a solution which is singular at is called a Legendre function of the second kind. If is an integer, the function of the first kind reduces to a polynomial known as theLegendre polynomial.

The Legendre differential equation can be solved using the Frobenius method by making a series expansion with ,(4)

(5)

(6)

Plugging in,(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

so each term must vanish and(15)

(16)

(17)

Therefore,(18)

(19)

(20)

(21)

(22)

so the even solution is(23)

Similarly, the odd solution is(24)

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 then given by the Legendre polynomials(25)

(26)

where is chosen so as to yield the normalization and is ahypergeometric function.

The associated Legendre differential equation is(27)

which can be written(28)

(Abramowitz and Stegun 1972; Zwillinger 1997, p. 124). The solutions to this equation are called the associated Legendre polynomials (if is an integer), or associated Legendre functions of the first kind (if is not an integer). The complete solution is(29)

where is a Legendre function of the second kind.

The associated Legendre differential equation is often written in a form obtained by setting . Plugging the identities(30)

(31)

(32)

(33)

into (◇) then gives(34)

(35)