Polynomial, Legendre

The powers of the variable x (1, x, x2. xa.) are not orthogonal functions over a unique interval. However, particular sets of polynomials present the orthogonality property. A simple and useful example is that of Legendre polynomials. Let us choose over the range [—1, +1] the first two polynomials 

Because of a different normalization Pf, Pf , the coefficients of the parentheses are not identical for the Gram-Schmidt orthogonalization and for the recursion formula. 

Setting the term containing 0 to a constant — m, we obtain the familiar differential equation 

Separation of variables in Laplace s equation leads to an ODE for the function 0(0)  

With a choice of constant such that Pt )= 1, the Legendre polynomials can be defined by Rodrigues formula  

Reverting to the original variable 9, the first few Legendre polynomials are 

An alternative generating function involves Bessel function of order zero  

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The interaction energy can be written as an expansion employing Wigner rotation matrices and spherical hamionics of the angles [28, 130], As a simple example, the interaction between an atom and a diatomic molecule can be expanded hr Legendre polynomials as... 

It is convenient to expand7 (0) in a basis of Legendre polynomials / (cos 0) (as these define the natural... 

Figure Bl.12.11. Angular variation of the second- and fourth-rank Legendre polynomials.

Wlien expanded as a series of Legendre polynomials /Jj (cos 0), tire differential cross section has the following form... 

The functions P " are associated Legendre polynomials of order m and degree I, and are associated Laguerre polynomials of degree (v — l)/2 in... 

These funetions are ealled Assoeiated Legendre polynomials, and they eonstitute the solutions to the 0 problem for non-zero m values. 

The functions are the associated Legendre polynomials of which a few are given in Table 1.1. They are independent of Z, the nuclear charge number, and therefore are the same for all one-electron atoms. 

We discuss the rotational dynamics of water molecules in terms of the time correlation functions, Ciit) = (P [cos 0 (it)]) (/ = 1, 2), where Pi is the /th Legendre polynomial, cos 0 (it) = U (0) U (it), u [, Is a unit vector along the water dipole (HOH bisector), and U2 is a unit vector along an OH bond. Infrared spectroscopy probes Ci(it), and deuterium NMR probes According to the Debye model (Brownian rotational motion), both... 

Fowlkes and Robinson [90] calculated the effects of the multipoles on the force of attraction by defining an interaction potential (p(r,6) and expanding that potential in terms of the Legendre polynomials... 

FIG. 5 (a) Fraction of adsorbed monomers (i.e., those with z-coordinate less than 6) vs ejk T for four different chain lengths, (b) The same for the second Legendre polynomial P2 cosd). (c) Scaling plot of and P2 cos9) vs distance from the adsorption threshold, using = —1.9 [13]. 

For each choice of n (the number of points), the w, and the n zeros ( ) of the nth degree Legendre polynomial must be determined by requiring that the approximation be exact for polynomials of degree less than 2n + 1. These have been determined for n = 2 through 95 and an abbreviated table for some n is given in Table 1-16. The interval - 1 < < I is transformed onto the interval a < X < b by calculating for each x, (k = 1,. . ., n)... 

The spherical harmonics are defined in terms of the associated Legendre polynomials, of variable cos 6, and exponential functions in... 

We may solve for the electron distribution function by expanding it in Legendre polynomials in cos 6 (where v = (v,6,Fourier series in cot we shall use here only the first-order terms ... 

Here u is a unit vector oriented along the rotational symmetry axis, while in a spherical molecule it is an arbitrary vector rigidly connected to the molecular frame. The scalar product u(t) (0) is cos 0(t) in classical theory, where 6(t) is the angle of u reorientation with respect to its initial position. It can be easily seen that both orientational correlation functions are the average values of the corresponding Legendre polynomials ... 

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

At the simplest level the orientational correlation of molecular pairs can be characterised by the averages of the even Legendre polynomials Pl(cos J ij) where is the angle between the symmetry axes of molecules i and j separated by a distance r. This correlation coefficient is denoted by... 

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