Orthogonality generalized spherical functions

The orientation is not strictly identical for all structural units and is rather spread over a certain statistical distribution. The distribution of orientation can be fully described by a mathematical function, N(6, q>, >//), the so-called ODF. Based on the theory of orthogonal polynomials, Roe and Krigbaum [1,2] have shown that N(6, generalized spherical harmonics that form a complete set of orthogonal functions, so that... 

In general case Eqs. (4.60) and (4.61) present infinite sets of the five-term (pentadiagonal) recurrence relations with respect to the index l. In certain special cases (t - 0 or a - 0), they reduce to three-term (tridiagonal) recurrence relations. In this section the sweep procedure for solving such relations is described. This method, also known as the Thomas algorithm, is widely used for recurrence relations entailed by the finite-difference approximation in the solution of differential equations (e.g., see Ref. 61). In our case, however, the recurrence relation follows from the exact expansion (4.60) of the distribution function in the basis of orthogonal spherical functions and free of any seal of proximity, inherent to finite-difference method. Moreover, in our case, as explained below, the sweep method provides the numerical representation of the exact solution of the recurrence relations. 

The expansion in Legendre polynomials or more generally spherical harmonics is chosen because they are orthogonal functions. The coefficients Ul in the expansion can be obtained by multiplying both sides of Eq. (24) by Pl(cos0) and integrating over 0, with the result ... 

The vector spherical harmonics YjtM form an orthogonal system. The state of the photon with definite values of j and M is described by a wave function which in general is a linear combination of three vector spherical harmonics... 

Implementation in the Rietveld method of the general representation of the texture by symmetrized spherical harmonics made possible a robust texture correction in the structure refinement and transformed the Rietveld method into a powerful tool for a quantitative determination of the texture itself. The robustness of the texture correction is a direct consequence of the fact that the symmetrized spherical harmonics are orthogonal functions. 

In general, for 2D flows, 3 can be identified with a Cartesian variable z, orthogonal to the plane of motion, and /13 = 1. However, for axisymmetric flows, c/3 represents the azimuthal angle

spherical coordinates, for example,... 

The most commonly used approach to the problem is to expand the correlation functions and their Fourier transforms in a series of orthogonal functions, usually the spherical harmonics. This approach was pioneered by Chen and Steele in the case of the Percus-Yevick approximation for hard diatomic fluids. More recently, the approach has been generalized to arbitrary... 

Considering the general null-field equation (2.4), we restrict r to lie on a spherical surface enclosed in D expand the incident field and the dyad gl in terms of regular vector spherical wave functions (cf. (1.25), (B.21) and (B.22)), and use the orthogonality of the vector spherical wave functions on spherical surfaces to obtain... 

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