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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 Pν(x) of the first kind (the restriction is usually added that Pν(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.

Def. Generating function. A function F that, through its representation by means of an infinite series of some sort, gives rise to a certain sequence of constants or functions as coefficients in the series.

James & James. Mathematics Dictionary

Generating function for Legendre polynomials

Recurrence formulas for Legendre polynomials

Orthogonality of Legendre polynomials

Because of 9), Pm(x) and Pn(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 Qn(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 Qn(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 Pn(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 Qn(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