Legendre polynomials analysis

From the uniqueness of the solution (Fig. 2.6j), it follows that M has to he an integer. The solution exists if 7 = 0,1,2,3, and from the analysis of the associate Legendre polynomials, it follows that M cannot exceed 7 because otherwise F = 0. The energy levels are given by... 

Racah s motivation for extending the vector analysis to tensors was his need to cope with matrix elements of the Coulomb interaction in a more systematic way than that provided by Slater (1929). The definition of a Legendre polynomial yields... 

For an axially symmetric system, only even series of Legendre polynomials are retained. A full description of the segmental orientation function requires that the individual component be defined. The coefficients (7s, characteristic of the distribution fimction, represent the contribution from each Legendre polynomials. The second-order coefficient is the well-known Herman orientation function (201) and is generally the only one used for structural analysis in pol5uners. 

The Galerkin-style analysis used here involves the expansion of the temperature in a truncated series of Legendre polynomials. Satisfaction is required of as many moments of the energy equation as there are unknowns in this series. The ensuing partial differential equations for the Legendre components are then solved. In the present treatment, only two unknoMi conqtonents are used. And for these it Is feasible to carry out explicit integration, as follows ... 

Tremblay G, Legendre P, Doyon J-F, Verdon R, Schetagne R. 1998. The use of polynomial regression analysis with indicator variables for interpretation of mercury in fish data. Biogeochemistry 40 189-201. 

A detailed analysis of polynomial approximation using collocation at Legendre roots by de Boor (1978) shows that the global error e(t) = Z(t) — z +i(t) satisfies the relation... 

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