Function spaces Legendre polynomials

P (z) are the Legendre polynomials [51] which now constitute the appropriate basis set), Eq. (132) may be solved to yield the corresponding results for rotation in space, namely, the aftereffect function [Eq. (123)] and the complex susceptibility [Eq. (11)], with x and Xo from Eqs. (81) and (84), respectively. Apparently as in normal diffusion, the results differ from the corresponding two-dimensional analogs only by a factor 2/3 in Xo and the appropriate definition of the Debye relaxation time. 

In this expansion, the dependence of the velocity distribution F U, vjv, z, t) on the direction v/v is fixed by the Legendre polynomials P v./v). Thus, averages with respect to the angle space v/v over the velocity distribution and appropriate weight functions can be performed. For example, with dv = dv d(v/v), the angle space averages over the velocity distribution F and over the product of F and F, yield according to (5) and (7) the expressions... 

The quantity 6 is the width parameter of the GWP, and Rq and ko correspond to the location of its maximum in the coordinate and momentum space, respectively. The functions vj r) along with the normalized Legendre polynomials (Pj(cos7)) represent the ro-vibrational eigenfunction corresponding to a (v,j) state of the H2 molecule. The functions vj r) are obtained by solving the eigenvalue equation of the free H2 molecule ... 

It can be more useful to use spherical co-ordinates [3] (see FIGURE 2). Any cylindrically symmetrical function with a centre of inversion can be developed into a series of even-order Legendre polynomials P2n, in a fashion similar to the way a periodic function is depicted by a Fourier series. The scattered intensity distribution over the reciprocal space can thus be represented as... 

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