Design of a Mathematica package to develop the product of some spherical functions as linear combinations of themselves. Issue 5 (7th August 2019) - Forner Gumbau, Manuel; Barreda Rochera, Miguel; López Ortí, José Antonio
Abstract : Results obtained when solving problems about atomic physics and about the potential theory show an almost spherical symmetry. These solutions can only be expressed in an exact form on a few occasions, so turning to approximations of these solutions through spherical functions becomes necessary. To achieve these approximations, it is necessary to obtain the product of two or more spherical functions as a linear combination of them. In this work, some formulas expressing the products of Legendre polynomials, associated Legendre functions and spherical harmonics, as linear combination of themselves, are presented. To do so, the bases for the algebraic manipulations of the previous products are first established. Subsequently, analytical developments with exact coefficients of those products are obtained. A software package has also been programmed with Mathematica program to obtain, in a simple and practical way, the aforementioned developments.