Legendre polynomials

The expressions of spherical functions and their derivatives in a Cartesian coordinate system are obtained.  In opposite to the representation of polynomials in a spherical coordinate system, the derived recurrence relations make it possible to use them in the description of physical processes, and the obtained formulae for derivatives of spherical functions within the sphere allow obtaining the solutions to the problems of mathematical physics for spherical bodies in a Cartesian coordinate system.  This approach has its advantages precisely in the applied problems.  For example, for  the d

The methods of calculation of gravity potential V and potential energy E for ellipsoidal planet for existing one-dimensional mass distribution were not worked out. That’s why now the derivation of formulas for simultaneous calculation of density mass’s distribution and potential and energy for the ellipsoidal body is actual.

This paper presents investigation of the image's possibility of distribution of lumply-continuous functions with are presented by Legendre polynomials and practical realization of this technique and methods for its improving were investigated.