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- Title
ON HYPERSHERICAL LEGENDRE POLYNOMIALS AND HIGHER DIMENSIONAL MULTIPOLE EXPANSIONS.
- Authors
CAMPOS, L. M. B. C.; CUNHA, F. S. R. P.
- Abstract
The Green function for the Laplace operator in N dimensions isobtained. It is used as the generating function for hyperspherical Legendre polynomials, that extend to N dimensions the original Legendre polynomials in three-dimensional space. The main properties, such as particular values,explicit coeffiients of expansion in powers, ordinary differential equation and recurrence and differentiation formulas are extended from the original Legendre polynomials in three-dimensional space to the hyperspherical Legendre polynomialsin any dimension. The hyperspherical Legendre polynomials are used together with hyperspherical and hypercylindrical coordinates to specify N dimensional monopoles, dipoles, quadropoles, octupoles and multipoles of anyorder. As a further application, the circle and sphere theorems involving the reciprocal point are extended to an hypersphere theorem: the latter specifies the effect of inserting in an uniform field an hypersphere with either tangential or orthogonal boundary condition at the surface.
- Subjects
HYPERSPHERICAL method; GREEN'S functions; LEGENDRE'S functions; LAPLACE'S equation; POLYNOMIALS
- Publication
Journal of Inequalities & Special Functions, 2012, Vol 3, Issue 3, p1
- ISSN
2217-4303
- Publication type
Article