Legendre functions of the first and second kind. Legendre differential equation. Legendre polynomials. Recurrence formulas. Generating function. Associated Legendre functions

Legendre differential equation. The equation

1)        (1 - x²)y" - 2xy' + ν(ν - 1)y = 0                    ν real

Solutions of this equation are called Legendre functions of order ν.

The Legendre equation is a special case of the associated Legendre equation

Legendre polynomials. Consider the case of 1) above when ν = n, a non-negative integer i.e. the equation

2)        (1 - x²)y" - 2xy' + n(n - 1)y = 0                   n = 0, 1, 2, 3, .....

The only solutions, with continuous first derivative for all x in the interval [-1, 1], of 2) are the Legendre polynomials, P₀(x), P₁(x), P₂(x), .... . However, if we allow y(x) to become infinite at the end-points, then the equation is also satisfied by functions Q₀(x), Q₁(x), Q₂(x), .... called Legendre functions of the second kind. For the case when the constant ν is real, the solutions are Legendre functions P_ν(x) of the first kind (the restriction is usually added that P_ν(1) =1), which are finite throughout the interval.

The Legendre polynomials are given by Rodrigue’s formula

The first few Legendre polynomials are:

If we set x = cos θ, the above can be expressed in terms of multiple angles as

It is in this form that they naturally arise in problems of temperature, potential, etc. for a sphere.

Generating function for Legendre polynomials

Recurrence formulas for Legendre polynomials 

Orthogonality of Legendre polynomials

Because of 9), P_m(x) and P_n(x) are called orthogonal in the interval -1 x 1.

Orthogonal series of Legendre polynomials

            f(x) = A₀P₀(x) + A₁P₁(x) + A₂P₂(x) + ..........

where

Some formulas involving Legendre polynomials

where C is a simple closed curve having x as interior point.

General solution of Legendre’s equation. The general solution of Legendre’s equation is

where

These series converge for -1 < x < 1.

If ν is a non-negative integer ν = n = 0, 1, 2, ...... one of the series 20) or 21) terminates. In such cases,

where

The non-terminating series with a suitable multiplicative constant is denoted by Q_n(x) and is called Legendre’s function of the second kind of order n.

Legendre function of the second kind of order n. Legendre’s function of the second kind of order n is defined for non-negative integral values of ν = n as

First several Legendre functions of the second kind

The functions Q_n(x) satisfy recurrence formulas exactly analogous to 4) - 8).

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Associated Legendre functions

Legendre’s associated differential equation. Legendre’s associated differential equation is

Solutions of this equation are called associated Legendre functions. We will restrict ourselves to the important case where m and n are non-negative integers.

Associated Legendre functions of the first kind. The associated Legendre functions of the first kind are given by

where P_n(x) are Legendre polynomials.

The following hold:

First several associated Legendre functions of the first kind

Generating function for

Recurrence formulas

Orthogonality of

Orthogonal series

where

Associated Legendre functions of the second kind. The functions

where Q_n(x) are Legendre functions of the second kind.

These functions are unbounded at x = 1, whereas are bounded at x = 1.

The functions satisfy the same recurrence relations as [as given in 37) and 38) above].

General solution of Legendre’s associated equation

References.

1. Spiegel. Mathematical Handbook of Formulas and Tables. (Schaum)

2. International Dictionary of Applied Mathematics

3. Wylie. Advanced Engineering Mathematics

4. James / James. Mathematics Dictionary