Cauchy formula

Formulas are derived for the calculation of the potential of bodies, which surface is a sphere or an ellipsoid, and the distribution function has a special form: a piecewise continuous one-dimensional function and a three-dimensional mass distribution.  For each of these cases, formulas to calculate both external and internal potentials are derived.  With their help, further the expressions are given for calculation of the potential (gravitational) energy of the masses of such bodies and their corresponding distributions.  For spherical bodies, the exact and approximate relations for determ

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