Legendre - DE, functions and polynomials of the first and second kind, associated Legendre functions

Legendre equation, function, and polynomial links (also contains properties and more).

http://mathworld.wolfram.com/LegendrePolynomial.html

http://mathworld.wolfram.com/LegendreDifferentialEquation.html

http://mathworld.wolfram.com/LegendreFunctionoftheFirstKind.html

http://mathworld.wolfram.com/LegendreFunctionoftheSecondKind.html

http://reference.wolfram.com/mathematica/tutorial/OrthogonalPolynomials.html

http://en.wikipedia.org/wiki/Legendre_polynomials

http://en.wikipedia.org/wiki/Legendre_function

The Legendre differential equation is of the form:

\[ \dfrac{\mathrm{d}}{\mathrm{d}x} \left[ \left( 1 - x^2 \right) \dfrac{\mathrm{d} F}{\mathrm{d}x} \right] + n \left( n + 1 \right) F \]

or

\[ \left( 1 - x^2 \right) F'' - 2 x F' + n \left( n + 1 \right) F = 0  \]

which has the general solution of

\[ C_1 P_n \left( x \right) + C_2 Q_n \left( x \right) \]

where \( P_n \left( x \right) \) is the Legendre function of the first kind, \( Q_n \left( x \right) \), and \( n \) is a non-negative integer which is also the degree of the Legendre polynomial.

A more general equation is the associated Legendre differential equation in the form of:

\[ \left( 1 - x^2 \right) F'' - 2 x F' + \left[ n \left( n + 1 \right) - \dfrac{m^2}{1 - x^2} \right] F = 0 \]

which has the general solution of

\[ C_1 P_n^m \left( x \right) + C_2 Q_n^m \left( x \right) \]

where \( P_n \left( x \right) \) is the associated Legendre function of the first kind, \( Q_n \left( x \right) \) is the associated Legendre function of the second kind, \( n \) is a non-negative integer, the degree of the Legendre polynomial, and \( m \) is also a non-negative integer which determines the degree of the Legendre functions/polynomials.

When the degree is zero, \( m = 0 \), the associated Legendre functions,  \( P_n^0 \left( x \right) \) and \( Q_n^0 \left( x \right) \), return the regular Legendre functions, \( P_n \left( x \right) \) and \( Q_n \left( x \right) \)

The Legendre DE can be solved by a power series expansion method also called the Frobenius method.

The difference between Legendre polynomials and functions is that when the \( n \)th-degree is an integer the Legendre function converges to a polynomial on the interval \( -1 \le x \le 1 \).

For associated Legendre functions of odd integers where \( m \le n \), the function contains \( \sqrt{1 - x^2} \) which demotes the function from a polynomial.

The Legendre equation usually arises in physics problems when the separation of variables method is applied the PDE Laplace's equation in spherical polar coordinates.

\[ \dfrac{1}{\sin \phi} \dfrac{\mathrm{d}}{\mathrm{d}\phi} \left( \sin \phi \dfrac{\mathrm{d} F}{\mathrm{d}\phi} \right) +  \left[ n \left( n + 1 \right) - \dfrac{m^2}{\sin^2 \phi} \right] F = 0 \]

Where \( \phi \) is the colatitudinal angle and the substitution \( x = \cos \phi \) recovers the previous Legendre equation version.