Legendre’s associated equation

Legendre s Associated Functions. We saw in example 1 of Chapter 1 that the solution of Laplace s equation in spherical polar coordinates reduces to the solution of the ordinary differential equation,... 

For m an integer the P/"(x) are polynomials. The polynomials P,°(x) are identical with the Legendre polynomials. The first few associated Legendre polynomials for x = cos 9 (a common form of Legendre s equation) are ... 

According to Roe s analyses, two kinds of orientation distribution functions given by equations (1) and (2) may be expanded into infinite series of normalized and generalized spherical harmonics as equations (4) and (5). Amnii) and Tlf(Cj) in equations (4) and (5) are the normalized associated Legendre s polynomials as given by equations (6) and (7). and in equations... 

The generating functions g " p, s) for the associated Legendre polynomials may be found from equation (E.l) by letting... 

Use of Associated Legendre Functions in Wave Mechanics. To illustrate the use of associated Legendre functions in wave mechanics, we shall consider one of the simplest problems in that subject — that of solving Sehrodinger s equation... 

In these coordinates Laplace s equation separates such functions of the form P ( )P. (7l)(g )m(J satisfy it where are the associated Legendre functions. Thus for this interior problem (i.e. inside the spheroid, not outside it) general solution of the linear problem is of the form... 

Spherical harmonics Y 9,(l>) are the angular contribution to the solution of Laplace s equation (or Helmholtz differential equation) in spherical coordinates (i.e. Eqs. (C.9) and (C.IO)). They are hence the product of the associated Legendre polynomial of cos0 and the general sine of the azimuth (/> ... 

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