Legendre equation

Equation 1.11 is known as the Legendre equation and it has solutions for integers m with as the reader may check in Appendix A. The solu-... 

The goal of this appendix is to prove that the restrictions of harmonic polynomials of degree f to the sphere do in fact correspond to the spherical harmonics of degree f. Recall that in Section 1.6 we used solutions to the Legendre equation (Equation 1.11) to dehne the spherical harmonics. In this appendix we construct bona hde solutions to the Legendre equation then we show that each of the span of the spherical harmonics of degree is precisely the set of restrictions of harmonic polynomials of degree f to the sphere. 

Proposition A.l The Legendre functions of Definition A.1 satisfy the Legendre equation. 

Proof. First we will show that the Legendre polynomial of degree satisfies the Legendre equation with m = 0. Then we will deduce that for any m =... 

Next we fix an integer m with 1 < m < and show that satisfies the Legendre equation (Equation A.l). Since the function P, o satisfies Equation A,2, we have... 

In the isotropic phase, differential Equation 2.106 is simplified to the Legendre equation with eigenfunctions, the Legendre polynomials Pn(cos9) and the eigenvalues n(n +1). 

Use the change of variables u(x) = 2x - I and show this leads to the usual form of Legendre equation given in Eq. 3.146. 

This equation can be transformed into a famous equation called the associated Legendre equation by the change of variables ... 

We do not discuss the associated Legendre equation, but give its solutions in Appendix F. The solutions are called associated Legendre functions and are derivatives of polynomials known as Legendre polynomials. 

For a solution of the associated Legendre equation to exist that obeys the relevant boundary conditions, it turns out that the constant K must be equal to /(/ -1-1) where / is a positive integer at least as large as the magnitude of m. There is one solution for each set of values of the two quantum numbers I and m ... 

Equation (F-46) is the same as the associated Legendre equation if K = lit + 1), where I is an integer that must be at least as large as m. The set of solutions is known as the associated Legendre functions, given for non-negative values of m by ... 

In order to make our shortcut with m = 0 pay off, we need to show that the associated Legendre polynomials satisfy the general Legendre equation. Consider P x) = 3(1 — y ). 

The purpose of this exercise is to show that the relatively simple solution of the m = 0 case of the Legendre equation can be extended to the general case for nonzero m values. The proof is given by Anderson [1] in two steps. First, define a function [f x) = (1 - x ) g(x)] and substitute it into the associated equation for nonzero values of m. 

This is an associated Legendre equation whose solution may be written... 

This equation is known as the associated Legendre equation, and the function Uy denoted by u = Pn( )j is called the associated Legendre polynomial of degree n and order m. From equations 4 36 and 4 37, we see that... 

The Linear Differential Equation of the Second Order, 48. The Legendre Polynomials, 62. The Associated Legendre Polynomials, 52. The General Solution of the Associated Legendre Equation, 53. The Functions 0j.r ( ) and 57. Recursion Formulae for the Legendre Polynomials, 59. The Hermite Polynomials, 60. The Laguerre Polynomials, 63. 

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