Legendre s polynomial

This is one of the main reasons why these functions play a very important role in solving boundary value problems. Also, between Legendre s polynomials of different order, there is a simple recursive relationship ... 

Let us notice that due to orthogonality of Legendre s polynomials many functions can be represented as a series, which is similar to Equation (1.162), and this fact is widely used in mathematical physics. Now, we will derive the differential equation, one of the solutions of which are Legendre s functions. 

A solution F = R(r) (single valued in (p only if is a Legendre s polynomial of the nth degree in which case R satisfies the equation,... 

Rodriques formula is useful to generate Legendre s polynomials for positive integers n... 

By making appropriate choices of , the linearly independent solutions can be reduced to polynomials (Legendre s polynomials). 

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... 

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 ... 

LEED, definition, 12-1 to 4 Legendre polynomials, A-83 to 85 Legendre s equation, A-46 to 56 Leptons, summary of properties, 11-1 to 55 Lifetime... 

There are a number of different kinds of orthogonal polynomials one can use, including continuous polynomials like Lagrange or Legendre polynomials [22], [21], and discrete ones, such as Hahn s polynomial [23]. The orthogonality property allows one to obtain the roots of the polynomial Xi, i =, 2,..., m — 1. Since orthogonal polynomials are also formed by linear combination of x or (for simplicity we can take the example of polynomials in x), Equation 2.30 can be rewritten at each collocation point in terms of new coefficients di... 

Let ) ( ) represents the (21 + l)th derivative of the (n + l)th Laguerre polynomial (20) and P7 (cos ) is Ferrers associated Legendre function of the first kind, of degree l and order m. Yim Zm thus constitutes a tesseral harmonic (21). The p s are in this form orthogonal and normalized, so that they fulfill the conditions... 

We next derive some recurrence relations for the Legendre polynomials. Differentiation of the generating function g p, s) with respect to s gives... 

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

Recurrence Relations for the Function Recurrence relations for the Legendre function of the second kind can be derived from Neumann s formula (18.1) anti the corresponding recurrence relations for the Legendre polynomials Pn(fi). From the recurrence relation (It.2) and Neumann s formula we have... 

The order parameter S is the orientational average of the second-order Legendre polynomial P2(a n) (n = the director), and if the orientational distribution function is approximated by the Onsager trial function, it can be related to the degree of orientation parameter ot by... 

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